It won't turn out perfect but it is a garage. *Insulation above the drywall & between the joist? *If the drywall has pulled away, there will most likely be debris between the drywall & the bottom face of the joists preventing oyu from getting drywall seated back tight to the joists.

No insullation above the garage ceiling so not much worry of debris.

How stuff do you have stored above? *Ceiling joists unless designed as attic floor joists aren't meant for storage. They're there to support the drywall ceiling.... have you over loaded the joistss/

Some boxes of kid's clothing and memorabillia, not much weight and
since I climb around up there on the joists, I think they're pretty
secure.

I'd check out whether or not you can "re-seat" the drywall, a couple (or a couple pair) *3rd Hand jacking struts would be helpful.

We just got a little lucky with that. When we tore out the kitchen
dropped ceiling there was a lot of 2X4 framework. We were about to
toss that today but instead cut it into pieces that would fit between
the joists. We ended up with about 50 pieces so that's a lot of in-
between support we can add in the spots where it's bad to help pull up
against.

If the end of the drywall are damaged & will not accept screws, consider some local joist sistering.

Joist sistering????

I prefer nails over drywall screws.....nails have bigger heads & with practice, *nailing isn't that bad. *I'd install fasteners at 6" o/c.

I agree on the nails, probably a really good place for that. I don't
mind if I ding the face a little hammering as I will have some mudding
to do anyway.

My suggestion is to attempt repair of the worst area first...if repair works there it will work everywhere.

Thanks Bob! Great suggestions and much appreciated!
Scott