I have NOOOOO idea. Maybe the washers in the faucet's shut-offs are acting up under different water pressure when the shower is turned on. I donno. Maybe 'high pressure' pushes on the washers just enough to prevent the faucet from dripping, but when the shower is turned on, the pressure drops and you get your leak.
At worst, rebuild the shut-offs for the hot and cold water in the bathroom sink. Good luck!
My best to ya and hope this helps.
Jay J -Moderator
PS: Don't just just 1 washer. Change them ALL. (You'll have the entire thing apart already and you may 'break' something else, so do them all ...