Looks like an old delta 600 series. Probably a 602. You have guessed exactly what you have to do to fix it, which should be easy enough.
1) Shut off the water to the shower.
2) Remove knob
3) Remove faceplate.
4) Unscrew the old diverter stem. (I seem to remember this being 7/8")
5) Repeat steps 1-4 in reverse order.
The new diverter stem is readily available, Lowe's item number 176062.
As for sealing around the tub spout, white silicone is the most common solution. get a good kitchen and bath type with microban in it to prevent mold buildup.