Furnace flames run for a minute then shut off
The high limit was indeed being reached. This was confirmed by watching the unit trip the high limit after the burners were turned on.
I had thought that the unit was getting enough air since my filter was new. It turned out that the secondary heat exchanger was packed full of matted dog hair and dust. I'll explain how I cleaned it.

Remove service panels, disconnect wires (orange stars), disconnect spark plug (pink star). Then remove the two screws marked with the blue stars to removed electrical box, gently set aside.

Disconnect the 3 wires attacked to the blower, be sure to note how they were connected, then remove the two screws that keep the blower from sliding out on the metal track (visible in the upper left). Then slide out blower.

Things should then look like this. The secondary heat exchanger is in the back and above the visible foil.

This is what the heat exchanger looked like, and what it looked like after a bit of cleaning.



I then vacuumed the rest of the heat exchanger as well as everything else to try and reduce future buildup. Everything is then put back in reverse order.
Thank you everyone for your help!