I broke the screw holding handle off and have about 3/4 of an inch still inside. How do I get it out?

About the only way at this point is to try drilling the rest of the screw out (or break the handle off the stem). No matter what you do, you'll need a new handle, new screw and, most likely, a new stem assembly... ;)

They also make screw extractors, but I don't think they will work in this case as the screw hole is too small to accommodate the screw extractor bits. You can Google them (screw extractor) and se what you think...

Try using a drill bit slightly larger than the screw and see how you make out.

Mark