I would say take something to keep you warm. I get cold easily too and I find that most everyone air conditions too much.

Both Bryce and the Grand Canyon are beautiful. They are both at elevations so they may be a bit cool. Maybe a good time for a souvenir jacket
I can't speak for Bryce, but the GC is down to 40 at night and only getting up to 70 during the day.

If you decide to do Bryce, you should check out Zion too, if you have not been there. They are close enough together that it is possible to see them both in one day if you do not dally. Bryce you look down on, Zion you look up.

You'll have to tell us all about your ventures when you return.