Assuming that is a pressure balanced valve (I think it is), if either the hot or cold water supply is shut off, the pressure balance spool valve will stop flow of all water (or at least most). SO, it sounds like it is doing what it is supposed to. It requires essentially equal pressure on the supply lines to allow your selected water temperature to pass through...shut one side off, the pressure balance spool valve slams shut the other side, stopping flow. The idea is you won't get scalded if the pressure drops in the supply (like when someone flushes a toilet, for example). This is not to be confused with adjusting the water temperature when both supplies are full on.
Moen will send you a replacement cartridge for free if you call them, so there's no need to buy one. You really want a 'real' one verses a knockoff made by some generic company. Some of them can be a real bear to remove...just do a search on remove moen cartridge, and you'll get loads of advice on how to do it.
If both supplies are on, and you can't adjust the temp to get any cold in, then the spool valve may be stuck at one end. I think this is separate from the temperature control valve, and replacing the cartridge won't solve your problem.