I had the same problem. I bought a new cartridge thinking that was it. I think my shower/bath faucet is a Delta 1600 or of the same design with the same parts. Replacing the cartridge did not help.
As a test I turned the cold water on with no cartridge and not a drop came out of the cold water inlet hole. The sink in the same bathroom worked fine. I figured something must have clogged the water line near the shower, or maybe the entire faucet assembly would have to be replace which would probably require removing tiles.
I learned about the pressure balancing spool (Moentrol valve). Apparently it can get corroded and then the piston can get pinned all the way to one side. I found the part on Amazon (I found at least two different styles so you have to make sure to get the correct part).
After I removed the old part I inspected it and noted that it requires quite a lot of work to remove or insert the piston inside of it due to corrosion (even after I tried cleaning it a bit by soaking in vinegar overnight). With the new part the piston inside will easily drop to the floor if you don't hold it in. When installing the new part I didn't do it right at first. I was worried about it being too tight, but I found I had to tighten it such that it "bottoms out" and the cap is flush against the housing.
Originally I was concerned about over tightening, but if you leave even a little gap the o-ring will escape under pressure. If that happens best to turn off the water again, remove, the part, reseat the o-ring, and do it again because if you try to tighten it after the o-ring "escapes" you will pinch the o-ring. Now about 24-hours and no drips, I kept the outer cover off so I could verify. Based on things I read I put a little silicone grease on each o-ring to try to make them last longer.
Maybe it's possible with some sanding or other measures I could have restored the old part, the key is that the piston has to be very free to move inside. Also, the five o-rings need to be in good shape (which mine were).
Also, I mentioned the piston can easily fall out if you tilt the new part, so be careful you don't accidently drop the piston behind the wall (luckily I didn't do that but I realized it could have easily happened when I was repeated removing the part due to not having it tight enough the first couple times because I did drop the piston into the bathtub each time). Also, make sure you have your drain plugged so you do not drop any screws down there.