Water dripping from shower head
The problem is that the "back pressure" at the tub spout is too high. Remove the tub spout and check/remove any debris. If there is no debris in the pipe or spout, then there are a couple solutions for this.
Adding a water flow restrictor to the pipe that runs up to the shower head might work. This would require cutting into the wall, though.
But I believe that the better solution (in my humble opinion) is to raise the height of the pipe that goes to the shower head, because it involves no painting.
Plumbing is all about gravity and pressure. If the pipe going up to the shower head is high enough, the evil back pressure from the tub spout will not be able to push the water to the top of the pipe, to overflow to the shower head. Adding a couple feet of pipe (into the attic maybe) might not require cutting dry wall, provided that you can cut through the top plate of the wall and reach and remove the FIP adapter (or drop ear fitting). You will first need to unscrew/remove the shower arm and probabaly use a stubby screw driver and small pipe cutter to detach the adapter or fitting. You will then need (in a difficult to reach area) to use a "sharkbite" coupling to add and extend the pipe. Add as much vertical pipe as necessary; generally, a few feet should do the trick. If you want to be sure it is high enough, have a helper turn on the water (after you have connected the extension). It should not overflow. Connect two elbows with a 2.5 inch nipple, to form a "U" shape. Attach the FIP or drop ear to a length of pipe- the same length as the extension pipe that is now connected to the "sharkbite" (minus the added length from the "sharkbite"). Finish connecting the pipes so that you form a "loop", up then down to the hole for the shower arm. Reattach your shower arm. Use 2x4s to build a support bracket for the loop and be sure to support the missing chunk of top plate (top of the wall).