Rmack1,
You may be on shore power, but you still need to check the house battery. There is a very good chance that the converter supplied with the unit is either failed or never was adequate to the task. Many light travel trailers are shipped with converter/chargers that will not manage this situation. Many of the furnace fans draw a lot of power and assuming that the house bank is OE, it is probably past its prime and cannot recover in the furnace cycle times.
Did you check that the battery water is at level? If it is not, that battery may well be toast by now, but topping it up is worth a try. Don't use campground tap water if you can get distilled or at least deionized.
I suggest that you get and carry even a cheapy of a meter to actually diagnose the situation. If you can't manage more, get a voltage readout that goes in a DC mousehole - lighter socket thing.
So, you have some choices right now.
=> If it used to work and now does not, replacing the battery may get you by. No Promise.
=> If it never worked, go get a good converter that can support the electric load of the heating system all by itself. This will be a good long term fix.
=> Get and electric blanket and leave the heat off until you are about to get out of bed.
Matt