'73 Door Handle went to crap
If they broke on removal, they must be stuck big time.
The only advise I can give is obvious,,, to do your best to get those holes started dead center.
It often helps to get the broken pieces of screws filed flat, this helps finding and keeping the drill bit dead center, but it is hard to do if the piece is still in place.
Start with a small bit and work your way bigger. With a little luck the broken pieces will come loose before you hit threads.
I have found that I do better drilling broken bolts out dead center when I absolutely have to do so.
The fact that I am opinionated does not presuppose that I am wrong......