Given the long term duration, yes I think you need to open the ceiling. In terms of the leak, it can be coming from the tub/shower valve....and almost any leak in that arena can be fixed just by removing the chrome trim pieces and replacing necessary parts. By the way, you can search for such a leak, also, by removing the handles and trim, then turning on the shower or tub and watch with a flashlight.
The leak also can be from the tub drain, or the overflow plate on the back of the tub. Both of those also involve gaskets which can be replaced from inside the tub.
Finally, the leak can be from a pipe or fitting under the tub, and that can only be fixed from below.
Don't be afraid to open the ceiling. You really do need to at a minimum get air circulation to dry out the wet areas. Patching drywall is not difficult at all. Getting it to LOOK GOOD takes a little practice, but you can get plenty of advice on that part, here and on many other forums. If the ceiling is in a high visibility area, you might just pay a good drywall guy to do that, especially if we can save you a ton of money and help you fix the plumbing yourself.
By the way, pictures of the valve, and anything from underneath when you open the ceiling, will be immensely helpful...helping us to help you.