It wasn't actually the shower head that was leaky, it was where the hose met the wall for the handheld unit. There was a gap between the connector pipe coming out of the wall and the connector piece on the hose. Every time you tried to turn on the handheld shower head (which has a hook on the wall so it functions like a normal shower head), you'd get sprayed in the midsection too. It wasn't the worst thing, but it made it difficult to soap up and properly shower. Up till now our solution was to just use the overhead rain shower instead.
I knew the fix would take a little research and a little creativity, and in the end that creativity took the form of putty: epoxy putty. A bit of putty to fill the gap and make sure the water goes where it should and things are all set. I took my first shower with the putty in place and everything held and is functioning.