Piper PA-28 pilot reported the engine began running rough and the mixture control lever would not move. The pilot was able to restore some power and returned to departure airport where they discovered post-flight that they had failed to notice the mixture control lock was on.

Narrative

Took off from ZZZ about XA:49 local time. Takeoff and climb northbound to practice airspeed calibrations for specific phases of flight. I climbed to my target altitude of 7;500. I after climbing; I reduced the RPM to 2400 cruise and 21 inches mp. The engine started to run very rough and I tried to change the mixture; but it was almost impossible to move. When I could move it; it jumped back shutting the engine off. I couldn't get it back in. The engine running very rough and the plane was loosing altitude. The roughness continued and I decided that I needed to land immediately; I tried to move the mixture lever but it was pretty much stuck. I decided maybe too quickly that I needed to do an emergency landing and land at ZZZ1 straight and headed for the airport. I had a hard time controlling the power and was busy running checks to see if I could find the problem. I didn't have ZZZ1 Tower frequency available to [make a] call and I couldn't let go of the yoke. I saw the airport in front of me and turned outbound to call them and [advise of the situation]; I had too much altitude as I was 1500 above their pattern altitude.On my turn outbound I found a power setting that would not shut the engine off; and I could control it; and I could stabilize the airplane. I'm not out of ZZZ1 airspace and figured that I would now try to get it as close as I could to my airport ZZZ. So I flew to ZZZ2 southbound because I had enough power to make it there if I needed to land. I made it to ZZZ2 and then felt confidence that I could get the plane back to ZZZ and land it without calling an emergency.I was able to land the plane at ZZZ with power settings I used for takeoff (Full prop/full mix). Landing was okay. After I taxied back to the flight school; when I shut the engine off; I found the mixture lock had been on from the previous pilot and I didn't exercise it because the engine was hot and I didn't want to prime a hot engine. So I didn't check the lock.I never called an emergency at ZZZ1 while I was in their airspace. For me at the moment the most important action for me was to fly the plane and try to determine if the engine was going to quit. When I found a power configuration that kept the engine going I was outside of their airspace and relieved and hoped it wouldn't repeat. By then I was over ZZZ2 and then made it to ZZZ.I told people at the flight school that I had engine trouble and was going to emergency land at ZZZ1. I wasn't sure if I should have called ZZZ1 tower afterwards to let them know why I was in their airspace.