Pilot reported a loss of aircraft control in IMC on approach after they mistakenly thought they had disconnected the autopilot. Pilot eventually disconnected the AP and completed the approach. Pilot also stated they overleaned the engine after landing; causing it to die while the airplane was not completely clear of the runway.

Narrative

I was making a flight with two pax to ZZZ. I departed VFR but when I was about 30 miles from ZZZ I decided I'd need an IFR approach to ZZZ because of clouds (OVC 019; tops 045). I asked for a clearance to the RNAV XX and was given direct ZZZZZ; maintain VFR; expect clearance at ZZZZZ1 (an IF but straight-in to the FAF). I accepted this. It became apparent I'd be in IMC before ZZZZZ1 so I advised ATC and was given a clearance for the approach; direct ZZZZZ; maintain 040. I had already loaded and briefed the approach; and the AP was flying in NAV mode.ATC then gave me a vector; I think for jet traffic. I switched the AP to TRK mode (it doesn't do HDG; just TRK) and attempted to enter a track that approximated the heading; but couldn't get the AP to accept it. I'm new to the airplane (45 hrs so far) and the AP; and have only flown one approach with the AP. I decided to disconnect the AP and hand-fly. I thought I had disconnected the AP (it has a knob that is used to both arm and disconnect the AP) but I had not; and I failed to verify on the status display that the AP was disconnected. The AP continued to fly the airplane while I attempted to hand-fly. The AP turned the airplane toward higher terrain. The airplane turned 180 degrees before I realized the AP was not disconnected and I was fighting the AP and the airplane was banked 30 degrees and perhaps 15 degrees nose-down. I turned off the power to the AP; recovered to straight-and-level; and then hand-flew us back onto the vector. The controller was calm; advising me of my heading and to turn to the heading he gave me. He then vectored me through the final approach course and back around to intercept the final approach course. This took probably three minutes. The controller switched me to tower and I keyed in what I thought was the tower freq from the approach plate but was actually ZZZ1 Center. I made the initial call-up and got a confusing response from the controller. About this time we broke out of IMC and I focused on ensuring we were configured to land. I tried tower again and got no response. I checked the plate again and realized the Center freq was similar to the tower freq and I'd keyed in the wrong freq. I put in the proper freq and was cleared to land. Landing was uneventful.I turned off onto the taxiway and stopped; and performed the after-landing flow. I over-leaned the engine and the engine died. Ground informed me that I had not crossed the hold-short line (it's in a non-standard place as the taxiway is at an angle) and then canceled the takeoff clearance of the aircraft holding short for departure. I re-started the engine; taxied clear. Tower informed me of a possible pilot deviation and gave me a number to call. I called the number and talked to the TRACON Supervisor about the incident.The primary factor in this incident was my relative unfamiliarity with the AP. I can certainly use the AP in VMC; but I need to know the AP better to rely on it in IMC. A contributing factor was my failure to confirm AP status after making a change to the AP. Another contributing factor is that the AP is independent - it does not take headings from the HSI not altitudes or barometer settings from the AI. A final contributing factor is that I experienced spatial disorientation and had not realized though my senses that the airplane was banked and nose-down. I do believe my experience as an instrument instructor led me to make the proper decision to disconnect the AP and hand-fly. My training to fly pitch attitude and keep the wings level; or banked standard-rate for a turn; prevented this from becoming a loss-of-control-in-IMC crash.I'm very experienced in type aircraft. This approach would've been routine for me in a type airplane.Finally; my failure to S-turn the airplane once on the ground to confirm I had cleared the hold-short lines was the primary factor in not clearing the runway (the C180 is a taildragger and the pilot cannot see the ground in front of the airplane. A contributing factor was my distraction from the difficulty of the approach and relief at being on the ground. As an instructor; I remind my students that landings after difficult approaches are often poor as you're so relieved at having gotten through the approach; and you cannot relax until after the landing. I need to revise that to include confirming being off the runway; and to use S-turns in a taildragger.