My single handle Moen shower drips every few seconds. I took the handle and chrome plate out. I can't figure out how to remove the cartridge.

Sounds like the cartridge needs to be replaced.
Click on; Installation Help / Animated Tutorial - Installing the Moen 1225 / 1200 Cartridge for detailed instructions. When the cartridge's out turn the water back on for a few seconds to flush out the suplies and opoen valve body. Good luck and thank you for rating my answer. Tom