Shower – How to prevent this bath faucet from leaking while the shower is running

You say you replaced the seats. The seat is where the usually rubber washer on the end of the stem seals against the back of the valve body. If you put your finger in the hole where the stem goes you should be able to feel the surface where sealing takes place. This surface can get scratched and allow leaks. In fact, the leak can make the scratch worse over time. Also, debris can become trapped in this space during repair and prevent sealing. So, make sure the seat is not scratched, it can usually be replaced if it is, and make sure no debris is left behind before closing up. Additionally, there is usually a washer made of paper fiber or hard plastic that sits between the valve body and the stem above the threads. If this washer is worn or missing leaking can occur at that point.

Finally, you say you replaced the seats and it got worse. It is possible the seats are not tight or are installed incorrectly, crooked, or cross threaded preventing the washer from seating. Check for that.

Good luck!

Diverting the leak up to the shower head using the diverter valve should not be a problem, but your assumption that fixing it will be "pretty easy" may not prove to be true. You are correct in that hot and cold supply to the tub/shower valve needs to be shut off. Sometimes apartment shower valves have built in shut-off valves called integral stops.

Sometimes apartments are equipped with individual remote shut-offs. Sometimes the whole building will need to be shut down. Removing the valves may require special tools like deep valve sockets.

The repair may be simple washer replacement or may require more complicated cartridge replacement.

I do recommend that you try it but make sure your landlord is cool with it in case something breaks, and have back up shower facilities available in case it ends up being out of service for a while.