Stereo power is not turned off by turning off the power switch on the unit. You need to add a switch into the 2 positive leads to the stereo to fully turn it off. Note that if you do this, you will lose your radio station memory and the equalizer will return to its default settings. This is no big deal to me, since the alternative is a dead battery if you can't leave your camper plugged in.
As the others pointed out, this is NOT the only thing that stays on whenever your battery is connected. The CO/propane alarm is always on. The stereo may go to a low power mode, or it may just stay on. The fridge power panel, thermostat, and water heater controls may also draw some power full time.
You either have to disconnect ALL the wires from one battery terminal or the other, or install a battery cut-off switch that cuts power to everything. Otherwise, just like your car/truck, you will have a dead battery in a couple of weeks.
For winter storage, it is safe to leave a fully charged, disconnected battery on the camper. But right now you don't have a fully charged battery, so you either need to plug in the camper for a couple of days to fully charge, or remove the battery and fully charge it before storing for the winter.
I learned the hard way about disconnecting batteries for winter storage - killing 2 batteries.
hope this helps
2019 Flagstaff T21TBHW A-frame
2008 Hyundai Entourage minivan
camping Colorado and adjacent states one weekend at a time