You've turned off your water to the bathroom and called a plumber or one tomorrow to mitigate additional damage, yes?

The plumber can test the pipes with canned air/applied pipe soap for bubbles to try to detect a leak via the access panel.

Older pipes often develop pinhole leaks. If you can cut a small hole in the wall near where you're hearing the dripping noise, maybe you can see what's going on. A plumber would do just that to find the leak.

(Op) I have plaster walls, not drywall-- but maybe it still won't be that bad. Our tub access panels are especially small and poorly located. Thanks.

There should be a removable access panel behind the tub in the adjacent bedroom that you can open to look for water as a place to start. Even if the leak is from the pipe that goes to the shower head inside the wall, having to remove a little drywall on the bedroom side won't break the bank.

I'm in an old house with original 1950's bathrooms. I can hear a new dripping sound in a bedroom wall next to the shower. I'm having visions of having to gut the wall to fix this leak. Any experience? I guess this could force us to do the bathroom Reno that we've been avoiding.