According to dictionaries, the correspondence with English is pretty good:
翻訳する and 訳す／訳する both mean "translate", to say or write something in one language that corresponds to another language. This is often written, but strictly speaking it doesn't have to be.
通訳する means "interpret", which is translating orally so that two or more people can communicate with one another. So technically, you're translating any time you interpret, but not the other way around.
Dictionaries mostly say 翻訳する and 訳す／訳する are synonyms in this sense.
You're right, though, that some people treat 訳す as "translate" and 翻訳する as specifically "translate writing". For example, on eigoTrans.com we find the following description:
当たり前のことだが、「通訳」は声を通じて言語を訳すのに対し、「翻訳」は文字によって言語を訳す。つまり、翻訳者は文章での表現力が、通訳者は音声での表現力が必要となってくる。
Here they use 訳す as a general term for translation, and they use 通訳 and 翻訳 for spoken and written translation, exactly as you suggested. But take a look at this definition for 通訳 from 明鏡国語事典, which uses 翻訳 in a very different way:
異なる言語を話す人の間に立って、話が通じるように双方のことばを翻訳して伝えること。また、その人。「英語を日本語に通訳する」「同時通訳」
If 翻訳する were always limited to written translation, this definition wouldn't make any sense―there's no written translation involved in interpreting. But I'm sure you understand this definition just fine anyway, so it's safe to say that 翻訳する can also refer to translation in general, not just written translation.
How do we reconcile this? Actually, it's pretty simple―it's based on context! We find the exact same distinction made in English in the same contexts. For example, Macmillan Dictionary defines interpret like this:
to translate what someone is saying in one language into another language so that someone else can understand it "I speak Spanish. Would you like me to interpret for you?"
It's clear that in this definition translate is used in a general sense. There's obviously no written translation involved in interpreting!
But you can find professionals who use the term translate specifically for writing:
Translation essentially consists of taking a text in one language and rendering it into another language. Did you catch that word “text”? The key characteristic of translation is the fact that it works in the medium of the written word.
This is the same situation we find with 翻訳 and 通訳 in Japanese! 翻訳 is often used specifically for written translation, but it's also possible for it to refer to translation in general. So to decide how 翻訳 is used, just like in English with translate, you'll need to pay attention to context.
And the same goes for your strategy of always saying 訳す. You'll want to choose which word you say based on context. You're right that you can use the word 訳す for both written and spoken translation, but if (for example) you want to talk specifically about interpreting, the most natural word is probably 通訳する, and you'll want to use that instead.
As a general rule, you should aim to use the same word a native speaker would use in the same context, and I'm afraid that means taking shortcuts like "always use 訳す" is counterproductive.
