Sentence a) is fine. You can omit the words "for example" if they are parenthetical, but that isn't always the case. Sometimes the words "For example start a new sentence:
I enjoy sport. For example, I play rugby every weekend.
You can give an example that is a separate utterance, but not a complete sentence. The example might just be one word:
People own many types of pets. For example, cats.
The second utterance isn't a sentence, but we understand it to mean "Cats are a type of pet." And with such an example we might punctuate with a comma "..., for example, cats."
And this is the case in your example. You could understand the example to be "Drinking too much was one of her poor choices". This sentence is the example, and the words "for example" are not a parenthetical insert into a sentence, but come at the start of a new utterance. If you remove the words "for example" then it's not clear what the meaning of "drinking too much" is, and so the English is not correct.