Cessna 182 pilot reported becoming distracted during single pilot operation in IMC on approach and entering an unusual attitude. The pilot took immediate actions to recover the aircraft and elected to do an approach into a nearby airport.

Narrative

In the morning; I started a VFR flight from ZZZ1 to ZZZ. I departed VFR knowing that the weather at ZZZ was marginal but possibly improving. As I approached ZZZ; I could see that that ceilings were not as forecast. Approaching; I decided to file a pop-up IFR flight plan with Center; who at the time; was monitoring my flight through flight following.As I approached ZZZ; I was handed off to Approach close to ZZZ2. At the time of hand off; ATC asked me what approach I would like and I choose the ILS for XXL; circle XXR. I choose this by default as I always will choose the ILS when able; due to the accuracy and the ability to provided a lower DA. My error in choosing this approach; was that this certain approach was in-operative at the time I requested. I was aware of the notam being ZZZ is my home airport. For reason's listed above; I choose this approach subconsciously making a critical error.I was allowed to continue on for the ILS at ZZZ. As I entered in to the nearby Class C airspace; I could see that the ceilings were possibly lower than what may have been originally forecast. As I approached north of the airport; I went to Ident the LOC frequency and did not get a tone. Although concerning; I thought maybe it was my location relative to the airport and had planned to check again as I got closer to the approach. As I entered the approach; ATC cleared me for the ILS XXL; circle XXR. As I was cleared for the approach; I went to test the identifier for the LOC once again. As I was doing this; I received communication from Approach that the ILS was inoperative at ZZZ. They changed my approach to the RNAV YYL Circle XXR.Being in full IMC; and as I already had the airplane set-up for the ILS; I struggled getting the information into the GPS and pulling the new approach plate so that I could properly brief the approach. As I approached the IF assigned; I was having issues putting the IF in the GPS. At this point; I had the AP (Autopilot) off as I always hand fly all approaches. As I blew the RNAV intercept; I attempted to turn back into it. Being distracted by my GPS; I inadvertently took my eyes off the instruments and did not have my AP engaged. At this point; I noticed a concerning sound in my headset; knowing it was wind caused by increased speed; looked at my VSI and seen a 1500 FPM descent. I was still in full IMC and took evasive actions to correct the attitude of the airplane. In addition to being in a dive; I was in a right banking turn. I was able to get the power pulled out; level the wings; and ultimately; able to stop the descent.During this time; ZZZ Approach broadcast an altitude alert. After getting the airplane stabilized; I reengaged the auto pilot and climbed to 3000 ft. At this point; I was given the option to reattempt the previous botched approach and I mentioned that I was not comfortable with the ceilings and would like to go to a VFR airport if possible. With nothing VFR available; I choose to set up for the RNAV XYL at ZZZ3.In the seconds that I experienced the incident; I lost close to 1200 ft in just a few seconds increasing my airspeed close to 180 kts. Everything leading up to the approach was comfortable; manageable; and for the most part; normal. I have re-ran the entire situation over and over in my mind; as well as reviewing ADSB data. After extensive review; I feel the point where things started to go south; was when I transitioned from the ILS to RNAV. I spent to much valuable time setting up the approach which led me not to take me eyes of my instruments; which is key to instrument flight. Knowing how to recover from an unusual attitude in IMC; ultimately saved my life!