PA-32 pilot reported autopilot and elevator trim malfunctions in climb as well as difficulty maintaining airspeed. The pilot disconnected the autopilot and pushed the yoke and trim wheel forward to regain control of the aircraft. ATC was informed of the altitude deviation and the pilot requested a diversion to a nearby airport.

Narrative

I departed ZZZ flight to ZZZ1 at XA:02 pm. The take-off and climb were normal. Then I turned to ZZZ VOR as directed while climbing to 4000 ft. MSL. I turned on the autopilot at about 1000 ft. MSL; but without altitude capture or hold. At 3200 ft. MSL; XA:08 PM; airspeed began to fall as I raised the nose to continue the climb. Briefly an airspeed and stall warning were heard. I reduced the rate of climb somewhat to keep airspeed above 80 kts; and we reached 4000 ft. MSL at XA:12 pm. As I tried to level the nose; I felt a strong back pull on the yoke and felt the trim wheel to my right at full back (up) position. I immediately turned off the autopilot and pushed both the trim wheel and yoke forward to gain airspeed that had fallen to about 78 kts. Noticing my suddenly decreasing altitude; ATC asked if I was having trouble holding altitude and I said 'yes'; and I recall that I referred to airspeed issues. I advised ATC asking for a nearby VFR airport. I was told that ZZZ2 with an 800 ft. ceiling is my closest airport and was given a heading and told to expect an ILS. My focus then became almost exclusively on airspeed and attitude; and then heading. At about 2000 ft. MSL in my descent; I confirmed that I had power in my engine. I was offered ZZZ3 (instead of ZZZ2) that was reporting an 1100 ft. ceiling; and I accepted it. At around 1600 ft. MSL I was able to raise the nose and put in power; and continued a normal; gradual descent below the 1100 ft. deck. I landed normally at ZZZ3 at XA:24 pm. At 4000 ft. MSL; given the airspeed warnings; I did not take time to trouble-shoot and seek possible reasons for the airspeed loss. I remain uncertain about this. Instead I opted to lower the nose; watch my attitude and descent rate and fly toward a nearby airport; ZZZ2 that was behind me. My main concern was controlling the airplane; so I left trouble-shooting for later. In my dialogue with ATC; I attributed the issue to a lack of engine power; but on reflection; I am not at all sure about that. I cannot point to any mechanical failure that caused this incident; and in a subsequent inspection; none were found.