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.
- 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.
- 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.
- 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!