Let us thank "Big Bang Theory" and the inimitable Christine Baranski character who has no trouble with "I need to urinate".

As for the language problem. There are two different scenarios which we are likely to run into: (1) A generic term for referring to someone respectfully who may have additional non-specified needs: 'Handicapped' may still work, 'Disabled' I think is in common use. (2) More specific terms where means of access or personal considerations are required: I just go straight to the facts of the matter with no attempt at euphemism, but I only do that with the people who are directly concerned. That way, privacy is maximized and anyone who wants to take issue with the wording has an opportunity to do so immediately and directly themselves.
And when I seek to be classless without being very specific, I use the word 'special' based on a manual I once saw at a fast-food franchise that referred to 'special' customers (In that case it was folks who wanted their money back).

In general, I respect individual privacy no matter what the issue is and no matter what I personally think of the individual. Causes way less regret and way less fear of the quiet loners amongst us. In Alaska we have a plethora.