2 questions. Where exactly does it drip from... and has there been a bead of clear silicone run around the exterior perimeter of the glass where it meets the cladding? And is the rot on the sash or the frame. (the sash is the part with the glass in it that cranks open, or would crank open if it operated.) Regardless, if you plan to replace the windows, it's a no brainer that you would get a new construction window that has a nailing fin. The thing you may want to rethink, if you have a brick perimeter, is resizing the windows and making them about 3-4 smaller. That way once you get the old window out, you can add a 2x4 or 2x6 around the entire perimeter, install the window, tape the fin, and then trim the perimeter with some trim. You'd probably want to add 1 1/2 of drywall and tape the perimeter of the drywall too. You don't have to remove brick to get the old window out, you will just have to destroy the frame, cutting it into smaller sections (reciprocating saw) until you can pull those sections off of the nailing fin. Then use a reciprocating saw to cut the nails behind the brick that are holding the nailing fin.

Normally those are hollow core doors and the rail is about 1.25 thick - then just air and cardboard slats. How wide is your door? Trying to determine exactly what you are working with. I've used toothpicks crammed in along with the screws before but drilling it out and gluing in a wood plug like PJ suggested is better.

Common problem at my house even without cutting the doors. The wood is so soft it does not hold the screws. Like suggested, I just drill 1/2 holes and glue a hardwood dowel into the doors. I use epoxy glue and never have the problem again.