BE-35 pilot reported fuel problems; being VFR in IMC and ultimately diverting to the nearest airport.

Narrative

Flying enroute from ZZZ1 to ZZZ2 at 7;500 cruise at 190 mph ground speed. Switched fuel tanks at specified usage time. Engine stumbled and started to die. Switched back to left main tank and engine came back. Checked fuel gauge and the right main indicated full so I switch back to the bad right tank and engine stumbled again. Instantly switched back to left main and called out mayday engine trouble. Was on VFR flight following with ZZZ Approach. The Controller directed me to the nearest airport and helped keep me calm. I am not instrument rated but I do have 15 hours of hood time. I was able to keep the engine running at idle as I entered the cloud layer at roughly 5;000 ft. I stayed in a standard rate turn to the right and circled the airport until I broke out at around 1;100 ft. The airport was on my right and I was able to put the gear down and flaps to make a good soft landing on the runway. Just as the airspeed started to bleed off the propeller came to a stop and I rolled to a stop halfway down the runway. Upon inspection the right main gas cap had popped open and all fuel was sucked out of that tank. The inner bladder had collapsed causing the fuel quantity gauge to show a full reading. Upon inspection of gas cap and comparing it to the other 3 caps It was determined the lock mechanism was too loose and may have come off in flight. At no point during the X hour flight did I notice any fuel smell or loss of power until switching to the right main tank. I have since fueled the aircraft and completed a ground run up on the tank in question and ops check OK.