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逃げる literally means "run away", which is natural to be intransitive even if you speak English. It also takes an argument with ～から, that corresponds to "from".

逃れる has a few different shades of meaning.

1. To keep away from a certain point where a (clear and present) danger exists. The "point" could mean a bomb, a burning building, or a man wielding his knife etc. In this sense, we use [point]-から逃れる.
2. To go out of (escape) a hostile area. "Area" means that it is a certain expanse of space, which could be an unstable country, a concrete jungle, or the state of poverty. In this case, we usually use [area]-を逃れる.
3. To avoid possible bad consequence in the future, which is not realized at this moment, but easily imaginable as a menace. Like arrest, public blame, or bankruptcy. It is also used in [consequence]-を逃れる.

～を逃げる sounds just wrong to me, although I can suppose that some people might use such a construction by analogy with 逃れる.

So your last sample phrase 役員を逃げた is pretty much ungrammatical, while 役員を逃れた is understandable in the way you evade being appointed executive.

Edit:
After some discussion in the chatroom, the transitive ～を逃げる might be a kind of older wording. It is very rare in BCCWJ (1976-) and have some appearances in 青空文庫 (out-of-copyright books). Modern speakers would not recognize this usage. We more naturally reword them in the present-day language.

役員を逃げる → 役員を回避する (?)
税そのものを逃げる → 税そのものを逃れる
網の目を逃げている → 網の目をかわしている
胸苦しさを逃げるために → 胸苦しさから逃げるために

Note that there is another "locational" を, which means "through" or "via", and does not make a verb transitive.

地下道を逃げるのよ！早く！
Run (away) through the basement (passage)! Hurry!

逃げる literally means "run away", which is natural to be intransitive even if you speak English. It also takes an argument with ～から, that corresponds to "from".

逃れる has a few different shades of meaning.

1. To keep away from a certain point where a (clear and present) danger exists. The "point" could mean a bomb, a burning building, or a man wielding his knife etc. In this sense, we use [point]-から逃れる.
2. To go out of (escape) a hostile area. "Area" means that it is a certain expanse of space, which could be an unstable country, a concrete jungle, or the state of poverty. In this case, we usually use [area]-を逃れる.
3. To avoid possible bad consequence in the future, which is not realized at this moment, but easily imaginable as a menace. Like arrest, public blame, or bankruptcy. It is also used in [consequence]-を逃れる.

～を逃げる sounds just wrong to me, although I can suppose that some people might use such a construction by analogy with 逃れる.

So your last sample phrase 役員を逃げた is pretty much ungrammatical, while 役員を逃れた is understandable in the way you evade being appointed executive.

Edit:
After some discussion in the chatroom, the transitive ～を逃げる might be a kind of older wording. It is very rare in BCCWJ (1976-) and have some appearances in 青空文庫 (out-of-copyright books). Modern speakers would not recognize this usage. We more naturally reword them in the present-day language.

役員を逃げる → 役員を回避する (?)
税そのものを逃げる → 税そのものを逃れる
網の目を逃げている → 網の目をかわしている
胸苦しさを逃げるために → 胸苦しさから逃げるために

Note that there is another "locational" を, which means "through" or "via", and does not make a verb transitive.

地下道を逃げるのよ！早く！
Run (away) through the basement (passage)! Hurry!

Edit 2:
I originally thought "transitivity" you mentioned meant whether the verb takes を or not, but it seems that you are concerned about the dictionary marking of 自/他.

In this sense, it is complicated because there are two kinds of criteria to judge it: whether it takes (non-location) を and whether it is paired with a more (in)transitive one. Those two agree in most cases, but there are always exceptions. For example, 着る and 休む^† takes を but paired with 着せる and 休める. In such cases, dictionaries' marking would vary. 逃げる and 逃れる both have counterparts 逃がす and 逃す, and that must be why you find them 自動詞 in your dictionary. But it does not necessarily explain the case alignment.

^{† They may be better understood as "reflexive ditransitive", one of whose object is the subject itself.}

逃げる literally means "run away", which is natural to be intransitive even if you speak English. It also takes an argument with ～から, that corresponds to "from".

逃れる has a few different shades of meaning.

1. To keep away from a certain point where a (clear and present) danger exists. The "point" could mean a bomb, a burning building, or a man wielding his knife etc. In this sense, we use [point]-から逃れる.
2. To go out of (escape) a hostile area. "Area" means that it is a certain expanse of space, which could be an unstable country, a concrete jungle, or the state of poverty. In this case, we usually use [area]-を逃れる.
3. To avoid possible bad consequence in the future, which is not realized at this moment, but easily imaginable as a menace. Like arrest, public blame, or bankruptcy. It is also used in [consequence]-を逃れる.

～を逃げる sounds just wrong to me, although I can suppose that some people might use such a construction by analogy with 逃れる.

So your last sample phrase 役員を逃げた is pretty much ungrammatical, while 役員を逃れた is understandable in the way you evade being appointed executive.

Edit:
After some discussion in the chatroom, the transitive ～を逃げる might be a kind of older wording. It is very rare in BCCWJ (1976-) and have some appearances in 青空文庫 (out-of-copyright books). Modern speakers would not recognize this usage. We more naturally reword them in the present-day language.

役員を逃げる → 役員を回避する (?)
税そのものを逃げる → 税そのものを逃れる
網の目を逃げている → 網の目をかわしている
胸苦しさを逃げるために → 胸苦しさから逃げるために

Note that there are another "locational" を, which means "through" or "via", and does not make a verb transitive.

地下道を逃げるのよ！早く！
Run (away) through the basement (passage)! Hurry!

逃げる literally means "run away", which is natural to be intransitive even if you speak English. It also takes an argument with ～から, that corresponds to "from".

逃れる has a few different shades of meaning.

1. To keep away from a certain point where a (clear and present) danger exists. The "point" could mean a bomb, a burning building, or a man wielding his knife etc. In this sense, we use [point]-から逃れる.
2. To go out of (escape) a hostile area. "Area" means that it is a certain expanse of space, which could be an unstable country, a concrete jungle, or the state of poverty. In this case, we usually use [area]-を逃れる.
3. To avoid possible bad consequence in the future, which is not realized at this moment, but easily imaginable as a menace. Like arrest, public blame, or bankruptcy. It is also used in [consequence]-を逃れる.

～を逃げる sounds just wrong to me, although I can suppose that some people might use such a construction by analogy with 逃れる.

So your last sample phrase 役員を逃げた is pretty much ungrammatical, while 役員を逃れた is understandable in the way you evade being appointed executive.

Edit:
After some discussion in the chatroom, the transitive ～を逃げる might be a kind of older wording. It is very rare in BCCWJ (1976-) and have some appearances in 青空文庫 (out-of-copyright books). Modern speakers would not recognize this usage. We more naturally reword them in the present-day language.

役員を逃げる → 役員を回避する (?)
税そのものを逃げる → 税そのものを逃れる
網の目を逃げている → 網の目をかわしている
胸苦しさを逃げるために → 胸苦しさから逃げるために

Note that there is another "locational" を, which means "through" or "via", and does not make a verb transitive.

地下道を逃げるのよ！早く！
Run (away) through the basement (passage)! Hurry!

逃げる literally means "run away", which is natural to be intransitive even if you speak English. It also takes an argument with ～から, that corresponds to "from".

逃れる has a few different shades of meaning.

1. To keep away from a certain point where a (clear and present) danger exists. The "point" could mean a bomb, a burning building, or a man wielding his knife etc. In this sense, we use [point]-から逃れる.
2. To go out of (escape) a hostile area. "Area" means that it is a certain expanse of space, which could be an unstable country, a concrete jungle, or the state of poverty. In this case, we usually use [area]-を逃れる.
3. To avoid possible bad consequence in the future, which is not realized at this moment, but easily imaginable as a menace. Like arrest, public blame, or bankruptcy. It is also used in [consequence]-を逃れる.

～を逃げる sounds just wrong to me, although I can suppose that some people might use such a construction by analogy with 逃れる.

So your last sample phrase 役員を逃げた is pretty much ungrammatical, while 役員を逃れた is understandable in the way you evade being appointed executive.

逃げる literally means "run away", which is natural to be intransitive even if you speak English. It also takes an argument with ～から, that corresponds to "from".

逃れる has a few different shades of meaning.

1. To keep away from a certain point where a (clear and present) danger exists. The "point" could mean a bomb, a burning building, or a man wielding his knife etc. In this sense, we use [point]-から逃れる.
2. To go out of (escape) a hostile area. "Area" means that it is a certain expanse of space, which could be an unstable country, a concrete jungle, or the state of poverty. In this case, we usually use [area]-を逃れる.
3. To avoid possible bad consequence in the future, which is not realized at this moment, but easily imaginable as a menace. Like arrest, public blame, or bankruptcy. It is also used in [consequence]-を逃れる.

～を逃げる sounds just wrong to me, although I can suppose that some people might use such a construction by analogy with 逃れる.

So your last sample phrase 役員を逃げた is pretty much ungrammatical, while 役員を逃れた is understandable in the way you evade being appointed executive.

Edit:
After some discussion in the chatroom, the transitive ～を逃げる might be a kind of older wording. It is very rare in BCCWJ (1976-) and have some appearances in 青空文庫 (out-of-copyright books). Modern speakers would not recognize this usage. We more naturally reword them in the present-day language.

役員を逃げる → 役員を回避する (?)
税そのものを逃げる → 税そのものを逃れる
網の目を逃げている → 網の目をかわしている
胸苦しさを逃げるために → 胸苦しさから逃げるために

Note that there are another "locational" を, which means "through" or "via", and does not make a verb transitive.

地下道を逃げるのよ！早く！
Run (away) through the basement (passage)! Hurry!