I think we are gaining, on the furnace and maybe the AC unit. Let's start with the furnace.
You show you have the Hydroflame 8531-III. This decoded down is an 85 series furnace, 30,000 BTU - generation III.
I do not know how electrically friendly you are on 12 volt DC circuits. If any of this is too much for you, stop. Only do any of this if you feel this is within your means.
This link will take to the 2003 tech manual for all of the Hydroflame furnaces. http://bryantrv.com/docs2/docs/hflamefurn04.pdf
To your wiring diagram. The images are too blurry to see but what I can see is that they gave you a variety of diagrams, several are not for your furnace. You have to find the right one amongst what they gave you.
This diagram in the upper right, is an AC control voltage furnace. I can see the 120vac transformer and the motor starting capacitor. I believe this is "not" what you have. So disregard it. I have not yet seen Sunline install a 120 VAC furnace unless someone else did.
This one is a lot closer, but I cannot read it all but this could be it.
In the link above I gave you on the tech manual, on PDF page 10 in the lower right corner is the wiring diagram for the 85 series - III with local flame sense feedback. It may be this one.
OR on the top of page 11, is the 85 series - III with remote flame sense feedback. It could be this one. I can't tell from what we can see on your pics.
The only difference between those 2 is the flame sensing circuit. The local sense uses the actual electrode to send a millivolt signal back to the ignition board to tell the flame is on, the remote sense one uses an actual probe to send a millivolt signal back to the ignition board. Right now the flame sense is not your issue so it may not even matter.
I do not know of a way to oil the motor bearings. I do not think they even have an oil port like some motors do. The blower motor is pretty buried in the heart of the furnace as there is 2 blower blades on the one motor. One for the inside blower fan to push the heat inside the camper, the other blade to blow the gas heat through the combustion chamber and back outside.
See page 16 of the PDF for a sequence of operation. And look at the wiring diagram on page 10 or 11 as mentioned above.
What we do know, on your furnace there is a relay that sort of runs the show. The T stat when it calls for heat energizes the relay and it's contacts close to start the motor. It also has a contact that when energized sends 12 volts to the sail switch. You jumping the wires in the camper simulates the T stat.
When the sail switch closes it sends 12 volts through a normally closed thermal disk switch which is a safety device as it opens up when the combustion chamber gets too hot and shuts down the ignition board. If the furnace is cool enough, the 12 volts is sent to the ignition board.
The ignition board then powers up the gas valve (you can hear a clunk of the valve opening) and it powers up the high voltage ignitor. (if you listen outside real close you should hear it sparking/clicking. The burner ignites and the flame sense feedback tells the ignition board the flame is on and everything keep burning until either the over temp switch opens or the T stat is satisfied.
Let's start with the simple things. Once the blower starts running, the sail should swing and close the switch. In about 15 to maybe 20 seconds of the sail switch closed, the gas valve will open and you hear a clunk. Followed by the igniter sparking.
From the point the igniter is sparking, there is about 6 seconds where the flame sense should tell the board flame is on. If no flame feedback it shuts down the gas and the ignition. I believe the older boards still did 3 tries to light and then goes into lock out. Meaning that after it did not fire off the 1st time, it will pause, allow the blower to run to purge the chamber, and then try again for 6 seconds. After the 3rd try of not flame feedback, it locks out on a flame fault safety until it is reset.
The blower on the older ones keeps running and you have to turn the T stat off, then back on to reset the system. In your case, untwist the wires wait a few seconds and jump them back together again.
Try and listen for the gas valve clunk and the sparking. Right now we do not know if the sail switch is being made, or the gas valve is opening or the igniter is firing. If you do not hear any gas valve clunk or sparking then the sail switch may have never made. If you hear sparking, then the sail switch made just the gas valve never opened. If you hear the gas valve clunk but no sparking, then the sail switch is working and the control board is partly working, but the ignitor may not be. OR there is a problem with the electrodes inside the combustion chamber.
If it comes that you hear no clunk or sparking at all, then it may be the control board is dead, or the sail switch never closed or maybe low voltage etc. You will have to sort that out. Here it gets more involved.
Start by checking the incoming voltage to the furnace. Voltage should be between 10.5 and 13.5 VDC at the furnace when the blower is running. There should be an on/off switch/circuit breaker at the furnace. Check the power between the switch and DC negative/ground for the proper voltage. Remember what the reading is with the blower running.
If that is good then by the wiring diagram, the white wire going into the ignition control board is the power feed from the sail switch/over temp switch. You will need to use a voltmeter on 12 VDC to see if power is ever reaching the PC board. Find the white wire and measure that to 12 volt DC negative. It should have power with the blower running. If no power the sail switch is not closed or the high temp switch has a problem.
Be careful. The above implies you know how to handle yourself around electricity. The igniter wire is high voltage like a spark plug on an engine. It will bite. If any of this you have not done like this before, stop. You can test all the clicks and clunks with the gas turned off. Right now you are trying to see if the gas valve ever clunked and the spark is ever sparking.
Hope this helps and let us know how you make out.