Are the caravan brakes properly released and the wheels free to turn?
I would certainly question the battery, whether its wearing out, but it could also be that its not being charged properly by the system in the caravan. You need to check the the voltage at the battery terminals with a good multimeter:-
If the battery is receiving enough voltage to charge the terminals on the battery should produce a reading of between 13.8 and 14.5V when the charger is turned on. If it doesn't reach that voltage, it means the charger isn't working.
Put the battery on charge for at least 24 hours, then Turn the charger and all the 12V systems off or disconnect the battery terminals and leave it to stand for at least 1 hour to settle. Then check the terminal voltage which if the battery has accepted a charge should be more than 12.6V. If its less than 12.6, it suggests the battery is not good.