My shower won't turn off and the handle feels a bit stiff (doesn't turn smoothly - but I have it turned all the way to right and the water is still flowing). The cartridge says Moen but that's all I know about it.
I feel comfortable with tools but have no plumbing experience.
I shut off the water to the bathroom for now so it's not a major emergency,
Is replacing the cart something I can do on my own, or am I asking for trouble?
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Maybe it's me but I would try to pull it myself. Sometimes they're difficult due to corrosion and scale buildup.
Once you have the cartridge puller and with the water supply turned off spray some Limeaway or other decalcifier on the cartridge. Let it sit for a bit then give it a shot. I haven't had to use a Moen puller but I doubt that if used according to instructions it could hurt the valve.
Although it was not a shower valve, my kitchen sink cartridge was real difficult to pull. Here is a video by Moen on YouTube on how to remove the shower cartridge you may have. It is very simple once you see how it is done.
The plumber may have not had the cartridge handle turned to the on position. But I would not let that guy touch my shower or anything I have again. To me it is a ploy for more work.