Heat Exchanger sporadically overheats

I have a 25 year old Amana furnace and a Lux Geo smart thermostat. The thermostat is powered off of the red wire (24VAC) and blue wire (Common). The thermostat runs on a schedule. If the temperature setting on the schedule is 70 degrees and I decide it is too cold and want to override the schedule to 72 (it overrides until the next schedule change time), I noticed the heat would come on but then within 30 seconds or so turn off and the setting on the thermostat would be back to 70 degrees.

I used an Arduino board as a data logger and sampled the 24VAC line at the thermostat and noticed that when the heat would come on, after 30 seconds or so the voltage would go away for about 20 seconds. This would reset the thermostat so when the 24VAC came back on, it would be back to 70 degrees. I also noticed that even when I wasn't overriding the schedule, the heat would come on, then shut off within 30 seconds then immediately come on again and run normally.

I have looked at the schematic (it has no real control board, just a bunch of relays and sensors controlling everything) and found that the only thing that can kill the 24VAC to the thermostat is the primary limit switch. I hooked up volt meters to the 24VAC going to the limit switch, the wire coming back from the limit switch (which goes on to the thermostat) and also a meter set to measure resistance placed across the limit switch. When the heat would come on the primary limit switch was opening after ~30 seconds and shutting down power to the thermostat and the entire relay board.

I assumed the primary limit switch was bad and removed it to test it. I used a thermocouple reader and a hot air gun and it seemed to work just fine. I then decided to install it again and also put the thermcouple in right beside of it. I found that the switch is working perfectly. When the heat first comes on, the pilot lights, the inducer fan comes on, the main burner ignites and even though the blower comes on, the heat exchanger will rapidly climb to 220F and of course the primary limit switch opens up and shuts everything down. But if I immediately make the heat come on again, the heat exchanger slowly climbs to about 140F and stays there for the duration of the heat cycle.

I know some sort of blockage or blower motor issue can cause the heat exchanger to overheat. But I can't imagine a blockage would be there when the heat first comes on, but then suddenly go away for a 30 minute heat cycle that immediately follows the failed heat cycle. The filter is clean and the evap coils were cleaned last year. So it makes me think it is some issue with the blower, but I can't think what would cause it to not run properly then on a second try run perfectly fine. Any ideas?