Our new 16 coachmen leprechaun 317sa few issues.

When my MH was about the same as yours had a similar problem. Removed the circuit board in furnace. Cleaned connectors on board with a clean pencil eraser. Put back together. Never had the problem again.
There was a mobile RV tech in park that we we’re staying in. This is what he suggested for a repair.

Look for a loose 12 volt connection to the furnace itself or to the thermostat. Try turning on the water heater or stove burners to get any air bubbles out of the propane lines. Make sure you're not low on propane. Poor house batteries/corroded connections can cause problems with furnace and appliance controls. Get a voltmeter and monitor condition of house batteries and charging from the converter charger.

When propane is low or almost out, my heater will act funny, similar to what you are describing. It will attempt to fire, the burner will kick off, but stop very quick then. After the third attempt and it's not getting the gas, it will stop trying and kick off completely. At that point I have to turn the thermostat into the "off" poisition and then back to the "on" position. It will attempt 3 more fires or kick-off's and then stop completely again. My camper is very consistent, and fortunately, I run only one propane tank at a time, keeping the second one turned off all the time, so I know when one is empty and they both are not empty at the same time.

When the battery is low, mine will fire, but the fan will run real slow, until it just putters out. Eventually, it just makes a little click and doesn't do anything. If I am plugged into shore power, even with a week battery, the furnace will still run OK. I got a new battery last Fall. With power plugged into shore power, the furnace ran fine. As soon as I unplugged form shore power, the battery had no juice, and everything in the camper (lights and all) just went extremely dim. The furnace literally shut down.

My suggestion? Check your propane before doing anything else.

First off, you have to see what your barreries are ar with a good volmeter. What you say is very symbolic of a weak connection not charging your batteries.
When the batteries are not charged, there is not enough to fill them, even when plugged in. In my case, the negative wire off the converter was so loose, the batteries could not get charged, so things did not work. Easily checked with a voltmeter, measure right at the converter output, and then at the battery. They should read the same.