Thinking about it, when I took the screws out, they were VERY hard to get out. They are definitely broken off at the ends - they are not all quite the same length and the ends are sheared off. Given how hard they are to get out, I suspect they are twisting off and breaking when I go to take them out.
I'm guessing I can't take them all out, put sealant in and then put them back since some of them are different lengths where they've broken off at slightly different places. (One of the longer ones would have trouble going into a whole with another piece broken off in it.) I could do them one at a time so that each screw half would match in the right hole, but that'll be time-consuming switching between a drill and sealant.
So while putting sealant in there sounds like a good idea, I'm not sure it's going to be straight-forward.