One of my two Atwood furnaces failed to ignite. All pointed to a sail switch.
The fan would kick on & run for about 30 seconds (to purge any gas in the burner chamber). I'd hear the gas solenoid kick in & I could smell gas (don't get your nose too close)but no "spark". The gas would shut off after about 5 seconds and the cycle would repeat three time.
I moved the board to the other furnace and had the exact same issue. I'm trying to decide if I should stick with the original board or go with a Dyno board.
Not saying your limit switch is good or bad. It's easy to throw parts at these furnaces till it's functional but that can cost some money.
Not a bad idea to pull the furnace, especially on an older rig. You might find a mouse nest or perhaps a large leak.
I can pull the entire guts out of my Atwood furnaces w/o pulling the whole furnace, you just have to know the procedure and what screws to remove.
Keep us posted.