C172 Pilot reported a rough running engine resulted in diverting to nearest airport to safe landing.

Narrative

Couldn't hold altitude and engine kept sputtering and wouldn't run smooth at all. No matter what settings of carb heat; throttle; and mixture. In the end full carb heat; full throttle; and full mixture were the settings that held power the best. The best power settings still couldn't hold level flight but did allow for a slow descent (200-300 fpm)Cessna 172. Flight scheduled to ZZZ and back for IFR time building for me and Safety Pilot. Preflight had 3 things I should have noticed and said no to flying that day for 2 of them. #1 - Extensive black exhaust soot covering the bottom left hand side of the fuselage. (Red flag)#2 - Missing screws on the cowling #3 - The plane owner did not inform me that this plane had been in an [priority handling event] exactly one week prior or and remediation or findings from that flight. Climb from ZZZ1 was 11 minutes to 6500. Cruise was 20 minutes at 6500. And descent was 13 minutes with engine never producing full power We both remarked how incredibly smooth the climb and cruise were. The Air was perfect for flying. We were at 6500 Running at 2300-2400 RPM cruising into a 13 kt. headwind for a ground speed of 75-77 kts. While I had my foggles on; I continually scanned all flight instruments and engine instruments. I never noticed the carb temperature reaching lower than 0 degrees Celsius. I Had foggles on my way to ZZZ2 with a Safety Pilot when the engine hiccuped. We both felt and heard it immediately. We had just switched frequencies from ZZZ center to ZZZ2 approach. Was about to call up and ask for ILS XXL into ZZZ. I initially thought mixture so pushed it rich and it smoothed up so I thought maybe he tapped the mix with his knee or something. Then 30 seconds later it coughed and almost started to die completely. I looked at carb temp and it was right at 0 degrees Celsius so I put on carb heat and it ran a bit rougher but seemed stronger. It was at this time my Safety Pilot reminded me I had foggles on and said 'you can get rid of those things'. I told my Safety Pilot 'I think we need to get to the ground it's not getting better; can you bring up Foreflight and find the nearest airport.' He said ZZZ3 and gave me bearing. So I pointed that way and called up Approach and [requested priority handling] saying engine failure can't hold altitude. At this point it was 3-4 minutes into it and I'd lost around 800 ft. of altitude. Approach said there's a couple airports due west 20/25 miles I can head to since ZZZ3 was IFR. I looked that way and it was forest and ZZZ3 direction was fields and I reply 'it looks clearer that way I'll try ZZZ3 but my glide says I can't make it and these fields look better.' So I point it that way and the engine is still sputtering and my Safety Pilot says 'what's the best glide speed? There's some good fields down here.' I say 'I've got power (1200-2000 rpm fluctuations) I'm not ready to glide yet let's try and make it closer' About 8 miles out I can see ZZZ3 and it's not IFR so I call up Approach and say I see it. Approach says do you want to switch frequencies and I say 'nope; my glide circle still shows I won't make it; I'd like to stay with you 'So I slowly descend (not on purpose) like 200-300 fpm until I get 4 miles out and feel comfortable I can make the runway so I pull up and slow up put my flaps in at 10 and tell Approach I have the runway made and he goes 'ok frequency change approved fly at your discretion'. And I reply 'frequency change approved but I'm flying at this plane's discretion thanks for all the help ' LOL Actually had to dump some altitude at the end and landed a bit long but I'm very confident that I couldn't have made it more than a couple miles past that airport. I think it was 15 miles away when the engine started having issues.Upon landing; there was even more exhaust soot all over the bottom of the plane. This was more pronounced than during our preflight. My Safety Pilot got a 99% on his PPL written [knowledge test] and the only thing he missed was what to squawk in [this situation]. I looked down a couple minutes into it and he squawked XXXX instead of XXXY. So he still hasn't learned. I have always wondered if I'd freeze or forget things in a situation like that and my training really kicked in. Safety Pilot said he never felt nervous or afraid and I sounded like a pro. On the inside I felt pretty calm and in troubleshooting and flying mode. But when I landed I got out of the plane and was shaking. They teach you power out emergencies and pitch for best glide but when I had power I was resisting that urge. I wanted to lift up and keep wings level and keep a bit more airspeed while I still had a prop running. Not sure if that was right but slowing down to 80 seemed wrong with some power. I was right on the edge of wanting to pitch for best glide. Its still under some question as to what the cause of the problem was; my CFII believes it was carb icing whereas others believe it was a faulty carb heater; faulty carb; or possibly a faulty induction system. It is still being investigated. If it were carb icing; then carb heat never fully recovered power during flight.