I replaced moen cartridge in kintchen single handle faucet and the faucet is running when the sprayer is on.

Replace the diverter valve. Different from the cartridge.

I agree with Harold also, Moen Faucets are life time warrantied I would suggest you to call them at 800-289 6636 gave your faucet model number explain your problem they will send the diverter or any other part needed for free, good luck.

John