I have uploaded the factory service/repair manual for the NT series furnace to Donna D. who is one of the gurus at large keeping the gears behind F.B.R.V.turning neatly.
I am told it will be one week before it is available on the F.B.R.V. site.
You will not see this available right now if you check. Give it a week or so to appear.
The service manual with all the test procedures with detailed explanations of the furnace step by step operations will be added for all to use. Once the manual is available and you have access to it, I will be glad to help you to start in trouble shooting the most likely cause of your issues with this furnace.
The unit is well worth the repair being a time tested design.
I have a pretty good idea now from reading your post where I would start, barring the exotic unknown bizarre problems, I suspect two small screws in the ground return path to the control board from the flame sensor, they are probably rusted in the mounting plate to the igniter assembly, its blocking the control board from seeing that the burner is lit...the control board is designed to shut down if it cannot prove ignition. In this model there are no pilots, its a direct spark ignition system...Yes two screws rusty can shut it down.
It will strike 3 times then lock off.
The manual has pics I can refer too.
Lets wait for the manual to appear, post when you have access to it, or email me thru the F.B.R.V. personal messages, We can open another thread or I will watch this one for a message you have it now.
Repair work is more a method than it is magic, a trail of clues that leads to a guess, the manual is well written from this point.
Good camping, Safe journeys
I think the Scamp is 1984. The furnace is under the gas stove, next to the refrigerator. I don't know how old the furnace is (if it's original). It's a Suburban NT 12 SE, called a Dynatrail. I live in California and as far as I know, the last owners were also in CA.