Pilot reported propeller RPM erratic and uncontrollable in cruise. The pilot diverted and made a precautionary landing.

Narrative

We (two passengers and I) departed ZZZ4 at about XA15 on Day 0; planning to fly from our home base; ZZZ4; to ZZZ3. I was piloting Aircraft X; a 1955 Cessna 180. The preflight and run up showed nothing abnormal; including three prop cycles from the run up RPM of 1;700 down to about 1;300 or 1;400 RPM. We departed ZZZ4 with radar flight following; and climbed to 9;500 ft. MSL across the state. During climb and cruise at 9;500 ft.; all engine parameters were normal; and the propeller control functioned normally when adjusted for a cruise RPM of 2;200. Approaching Location; I initiated a climb to 13;500 ft. at an indicated airspeed of about 90 MPH; with throttle wide open and prop RPM adjusted for 2;450 (cruise climb setting). The climb seemed normal; but as we leveled off at 13;500 ft.; about 5 or 10 miles east of ZZZ; the propeller RPM slowly increased above the red line of 2;600 RPM as the plane accelerated. Our JPI engine monitor flashed red as the red line was exceeded; so I immediately throttled back to a low power setting to keep the RPM below red line. I also immediately turned back toward the lower elevation; and away from the mountains; since propeller over-speed is sometimes caused by loss of engine oil pressure. I exercised the propeller control; and found that it had no effect on reducing or controlling propeller RPM. However; the engine oil pressure was normal and stable. I told ZZZ Approach about our propeller problem; and that I would descend for landing somewhere in the lower elevation area. We initiated a low-speed low-power descent; to keep the RPM below red line; and proceeded toward ZZZ1 which was soon within gliding range. During the descent; oil pressure and temperature remained steady and normal; so I decided to proceed to ZZZ2 airport; with a longer runway and where I hoped that maintenance services might be available. As we descended to ZZZ2; I considered my approach and landing; and not knowing the root cause of the problem; decided that I would like to have priority for landing; so that I wouldn't be vectored for traffic or possibly be instructed to go-around. So; I told ZZZ that I would like to [request priority handling]. ZZZ acknowledged this; and requested souls and fuel on board; and soon handed us over to ZZZ Tower. ZZZ Tower cleared us for a straight in approach to Runway XXR; but since we were still very high on the approach; I requested right downwind for XXR; and proceeded to lose altitude as we maneuvered onto the downwind. Approach and landing were normal; and Tower instructed us to taxi across Runway XXR to the south side of the airport; where GA Maintenance services were available.We tied down the airplane; and found an A&P mechanic; who inspected the prop control and verified that it was still connected to the governor control lever; implying that the governor itself had failed in some way. We left the plane at ZZZ2 and returned home by car. The mechanic has ordered a replacement prop governor; which is not yet installed as of this date.