I got another company to squeeze me in this morning. I zipped home, let the guy in, and gave him the run down. His conclusions pretty much matched up with mine. He hit all the reset switches I did and they all seemed to be in place.
See, all the limit switches are in series so if one blows you only know that one went down, not which one. One end plugs into the controller for the gas flow while the other plugs into another board along with several other wires. I considered this but gave it a pass for some reason. See all those wires go into one big plug. He detached that plug and plugged it back in place. It caused some related limit switch on the board to reset and everything came back on.
He charged me $70 which was their basic "yep, it's broke" fee. Or as the joke goes "the $70 ain't for flipping the switch. The $70 is for knowing which switch to flip."
Anyway, we're pretty sure the problem was initially caused by a clogged air filter. All the dust I've kicked up clogged it faster than normal. Plus I got a good filter. The filter that was there when I moved in could be smacked against a wall and replaced every few months. Even clogged it didn't really stop the air. So, it's a Home Depot trip tomorrow.