M20K pilot elected to divert to an intermediate airport after fuel indications did not reflect fuel calculations. Pilot reported some coordination issues with ATC regarding destination and alternate selections.

Narrative

Enroute to ZZZ1 from ZZZ; my fuel computations and fuel gauges were in disagreement. This airplane is new to me; so I decided to land short of ZZZ1 and refuel. I requested to change my destination to ZZZ2. I believe ZZZ1 center had just handed me off to ZZZ center. ZZZ2 was approximately 40 miles off my nose. The controller said roger; and gave me a descent from 10;000' to 4;000'. After leveling off at 4000; he then told me I would have to hold for at least 20 minutes to get into ZZZ2. I asked if I could go to ZZZ3 instead. He replied I would also have to hold at least 20 minutes to go into ZZZ3. While I was considering my options; another controller came on the frequency and said I could go to ZZZ4; which was about 40 miles behind me without holding. I agreed; and he gave me a turn to 120 degrees; which was almost 180 degrees from my present heading. I reset my destination and set up for the approach. As I was nearing the IAF for the RNAV Rwy XX approach; I asked if I could proceed direct to the fix to initiate the approach. The original controller replied with 'which approach would you like to shoot?' I replied the RNAV XX. Then the second controller came back with another 180 degree turn to go back to ZZZ2; which was now 50 miles behind me. I asked 'You want me to turn around and go back to ZZZ2 now after completely reprogramming my GPS/NAV for ZZZ4?'. He replied 'yes; we can get you in now.' I said 'Fine; but if I get delayed much more; I'm going to have to declare an emergency and land'. The second controller gave me a heading for the IAF to ZZZ2; and asked me to let him know when I was ready for the approach. I reprogrammed the GPS/NAV to ZZZ2 RNAV XY for the second time; and let him know I was ready for the approach. He [cleared] me for the approach and asked me to call the FAF inbound. When I reported the FAF inbound; He cleared me to unicom and asked that I include my down time when I closed out my IFR flight plan. I switched to unicom; landed and called ZZZ center's number on the ZZZ2 approach plate to cancel IFR and reported my down time.Four days later I got a call from the ZZZ5 airport manager that someone had asked me to call them. I called several times and left a message at the number provided without response.I'm not sure what I did that was wrong. I made the conservative call to land short when the difference between computed fuel and gauge readings became apparent. Had the controller informed me of the 'at least 20 minute hold' at ZZZ2 before descending me out of cruise; I had many options. However; after descending me; then directing multiple headings and destinations; my options were becoming limited. That was my intent when I said what I said. However; I did not declare an emergency. As it turned out; I still had an hour and a half of fuel remaining when I refueled; but I wasn't sure of that at the time. The airplane will be going into maintenance to recalibrate the fuel gauges before any more long trips are taken.