Every time I take a bath and drain the water I find water dripping from the ceiling below the tub. This doesn't happen when showering. What could the problem be? Leaking pipe or bad tub drain?

I don't think you have a drain problem, you have a mixing valve problem. When you shower the diverter sends the water up to the shower head, bathing the water comes out the spout. Remove the handle and the bezel of the mixing valve and inspect with a flashlight with the water running into the tub. You should see you leak. Come back and tell us what you see and several of us can help you fix it. Good luck, Bob