Turning the house batteries off would should shut down the fan (if the 110v is not hooked up) but I agree that is a Band-Aid and does not really solve the problem of why the fan runs when the furnace is in off mode or the temperature is set low enough so that there is no call for heat.
My Encounter coach has two thermostats, one in the hall and one in the bedroom, have you checked to see if there is more than one thermostat?
Normally after using the furnace and when the requested temperature has been reached, the heat function will shut down but the fan will run for a few minutes clearing out remaining heat and cooling the heating element. So... have you tried setting the heat control up, letting the furnace run for a few minutes and then adjusting the thermostat down to cycle the furnace and see if after a few minutes the fan stops? Doing this is pointless if you don't have the proper setting across the board. Unplug the 110v, set fan to auto, make sure the house batteries are "on", and check for that second thermostat and make sure it is off as well.
I don't know your experience level so I'll offer one other wild thought. Check all the ceiling fans and make sure none of them are running and are being confused with the blower on the furnace. (You did say air was coming from an outlet on a wall though.)
If that doesn't work it might be time for a call to the dealer, your son-in-law, or forest river.
Completely unrelated..... I am surprised your daughter and son-in-law agreed to this "test" as you called it. This is no doubt a huge investment for them and it takes some time to learn the systems, sort out problems, and figure out just what using an RV involves. I love my children, children in law. and all my grandchildren but I have been firm in my negative responses when asked if they could use the coach for a weekend, a concert, or a vacation. It's a lose lose. You can do damage through ignorance or lack of training and then no one will be happy, not my kids, not me. The first couple of discussions were painful for me, I hate to deny them anything but I didn't see this as a good idea under about any circumstances. They agreed after giving it some thought.
For what it's worth.
Gary and Cindy
Gary, Cindy, Reacher (Maine Coon), and Vivian (in spirit only now).
Retired: July 31st, 2013 and loving it.
2011 Coachmen Encounter 37 TZ pulling a 2012 Wrangler Unlimited Sahara. Many upgrades, additions, mods, and changes.