If you have a problem, tape your wrists until they heal, and then continue to tape them until you forget why you are taping them. In the meantime, take some anti inflammatories, take care of your wrists, and HEAL thyself.
If it's trashed, a proper taping is really the only good solution -- the elastic jobbies "support" but are really too flexible to prevent you from getting re-injured. Look around at all the "old guys" on the mat, including the shihans -- if they've trashed their wrists, they use good old surgical tape, possibly with an underwrap -- I don't think I've seen many with elastic bandages wrapped around, unless they taped over them, too.
Also, ask someone who knows how, to show you the proper way to tape, and have them explain it to you... there's more to it than just putting it on until you can't move the joints well. Good Luck!
PS - been there, done that, tried the elastic and it SUCKED. I'd rather cut off tape than deal with a stinky, stretchy, unhygienic overused elastic bandage. And the underwrap saves those little hairs that otherwise HURT when you rip off the tape.