What do I do if drywall screws won't penetrate the studs?
I suspect the studs are really hard old-growth wood


Lubricating screws can really help when driving them into wood. I've seen various lubricants suggested. For exterior projects I've used LM grease and petroleum jelly (vaseline) but I'm not sure if there are any problems using either in your situation.

The other common solution is to drill pilot holes to the expected depth of the screws, but this would be pretty tedious. I usually pick a drill size by holding screw and drill up to the light together and checking that the screw's threads are wider than the drill and that the drill is almost as wide as the core of the screw.