I just replaced the cartridge in a Moen single handle faucet which was leaking. Now that the new cartridge is in when I push the handle down to turn off the water it pushes back out and a small stream of water continues to run. If I put some pressure on the handle the water will shut back off, but as soon as I let go the stream resumes.

OK! I had the same problem with a new kitchen faucet I installed. The handle wouldn't stay up and slowly came down and shut the water off.
I called the company and was told that they lubricate the stem and "O" rings with silicon jel and after the water washers the lubricant off everything will return to normal. And it did.
Use it for a week or two band if it still gives you a problem we'll look into it farther. Patience is the operating word here/ Good luck, Tom