Leaky Shower Drain...Caulk or Replace?
Caulk is not a plumbing solution, and was never the right way to install a drain. (Plumber's putty is also on its way to the bin of history, having been supplanted by more reliable and less messy synthetic gaskets.) You need new gaskets or an entirely new assembly.

The only difficult part about tub plumbing is usually access. If you can get to the area in question, the rest is somewhat academic. Parts are generally fairly standardized and information abounds.