I would say, as in so many instances: it depends. The three words you singled out, pussy, ass, and bitch, have all been "taken over" by their more obscene meanings - that is, an average listener1 will either be confused, or at least temporarily distracted, if you use these words. For "pussy", I would substitute kitty; for "ass", I would use donkey; for "bitch"... unfortunately female dog is the shortest substitute I can think of.
Of course, these words still retain their original meanings, and it's silly to avoid them if you are writing for a purpose or for an audience where the original meaning is not only clear but central to the work. If you're a professional dog breeder, of course you use the word "bitch"; otherwise it becomes very hard to talk about breeding dogs. If you're writing a paper about Equus africanus, by all means call it the Wild African Ass; "donkey" would be not only inconvenient but wrong. If you are writing about the Salix discolor tree, call it a pussy willow! But in almost any other context, these words (and I'm sure we can all think of a few other, similar examples) will prove more of a distraction than a help.
That being said, people come up with obscene meanings for innocent words all the time, and if you try to avoid all words that could possibly be misinterpreted, you'll never get any writing done! Bone, for instance, is commonly used as a euphemism for "fuck" - but only a teenager will giggle if you talk about "the bones of the inner ear". Come is, of course, commonly used to mean ejaculate (both verb and noun)2, but it would be silly to avoid using such a common verb simply because some sixth-grader might be titillated.
1 At least in the United States; I can't speak for anyone else.
2 Come to think of it, ejaculate is another word you should probably avoid, even though it has a perfectly innocuous meaning as well as a sexual one.