Did the weakness and numbness start after your adjustment, or was it there before? What did the chiro advise? Can you tell whether it's coming from your neck or more from the shoulder joint/thoracic outlet? Does stretching help? Pinched nerves are tricky...

I guess if it was me I'd talk to the chiro, maybe get in for a massage at their office, do a little VERY LIGHT activity through a full range of motion to see what happens, and go from there. I'd include a little perch on the bike - which can stress the thoracic outlet - as well as a bit of a swim. If either one of those seemed to aggravate the problem, I'd probably pass on the tri. You definitely don't want to bruise the nerve, because that can take months to heal.

But that's just what I would do, NOT medical advice...

Anyway, if you feel you can do the tri without injuring it further, good luck!