Very common shower, and fairly easy repairs. First, don't forget to turn off the main water shutoff.
As for the broken screw, just get a dril with something like a 3/8" bit. Just obliterate the srew head. You will possibley damage the handle, but it probably needs to be replaced anyway, and they are cheap. Once you get the handle off, you may be able to remove the stub of the screw from the stem with vice grips, but again, if this shower has any age on it, you might want to just renew all the stems and all the trim. It is all readily available.
Anyway, after you get the handle off, the chrome trim flange wil also unscrew, usually just hand tight. Now, a special deep shower socket will remove the entire stem. NOTE: Before trying to unscrew the stem, open the valve most of the way. Same when reinstalling....dont try to tighten or loosen the stem if it is screwed in to the "off" position.
If the brass is in good shape, just replace the flat washer ( size is 3/8L ). Lubricate the stem shaft with plumbers grease; work the stem in and out a few times to distribute the grease. Put is back in. Notice that there is a flat fiber or plastic gasket where the body of the stem seats into the valve body in the wall. Make sure that is in place, and if it got nicked up, just replace it.
For the one that is leaking around the handle, this could be that flat gasket, or more likely it is the fat packing around the stem. Try tightening the smaller, outer, nut. TIghten ins 1/4 turn increments until there is no drip of water out there. If you cannot get it to seal without tightening so much you cant turn the handle, replace the stem packing, or better yet, just replace the whole stem, which should be $9 to $14.
One more thing: before putting it all back together, inside the valve body in the wall is a part called a seat....the flat washer squeezes against it. It is normally replaced at the same time as the washers. Now you need a seat wrench, which looks like an overgrown allen wrench.
From a previous thread, here is a typical breakdown, showing parts and the tools needed: http://www.terrylove.com/pp_3.htm