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Hi Luke, I'm Brian, an HVAC expert here at Just Answer and I'm here to help!
The fact that the furnace runs for so long before shutting down on high limit usually indicates either a weak high limit switch, or diminished airflow because of a dirty air filter (the most common reason for this), dirty blower wheel or dirty air conditioning a-coil. I would check for the airflow conditions first and if all is clean then I would replace the limit switch with a new one with the same ratings. I would also check the temperature rise of the furnace and compare it to the data plate on the furnace. The plate should have a 'Max Temp Rise' or the like. That is the temperature difference from entering to leaving air on the furnace. If the furnace is running at or near the max, then there is some kind of diminished airflow problem.
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Ok, it sounds like the high limit switch may be getting weak and tripping lower than it should. The way to double check that is to check the temperature rise by measuring temperature of the return air and the supply air and subtracting. Compare that to the "Max Temp Rise" on the furnace's data plate. If the temp rise is well under the max temp rise, then you most likely have a weak limit switch.