How can I fix a Moen faucet that is still leaking after replacing cartridge?
I contacted Moen directly with this same question and they let me know that they recommended a model 1225 replacement cartridge instead (that's what I get for trusting the guy at the hardware store to look up the matching part for me). Technically either one will work but apparently the plastic 1225 offers a tighter fit than the brass (and more expensive) 1220.

Moen also recommended cleaning out the inside of the valve with an old toothbrush or bottle brush and then flushing thoroughly before installing the new cartridge.

I will update this once I know if the new cartridge resolves my leak.

Update:
After the installation of the model 1225 cartridge, the faucet works without any leaks from the stem. With the addition of a new knob the whole fixture is working better than new.

The moral of the story here is, when in doubt, try to contact the vendor (and post your results!)