It does sound like the battery is the problem. If it is older, or if has been run completely down a few times, it probably won't hold a charge. I ran into this in the spring on our popup. When I put the battery on the charger, it would show about 80% and then shortly it would show a full charge.
Also - you mentioned your battery is going dead while traveling. You may want to check your trailer plug to see if you have 12 volts on the DC pin for charging. I had this issue with both Fords I have had. The factory does not install the fuse or the relay that powers that circuit and send voltage to your trailer. Brakes and trailer lights worked, but until I added the fuse and relay I didn't have power to charge. this seems to be true for several makes/models. Nothing worse than pulling in for 5 days of camping in park w/o elcectric and finding your battery dead! I was able to pull the cover on the truck fuse panel and the diagram showed the sockets for that circuit.
I would guess if the battery is original its time for a new one, and then check your tow vehicle for power. If the popup doesn't have a battery disconnect switch, pull the positive wire off the battery whenever you store it to keep it from going dead, and it should last a long time.
2005 F-150 FX4
2013 Rockwood Roo 23 IKSS