Yes the 13+ volts is a surface charge from the charger. A fully charged battery should be around 12.7v. Put it on the trickle charge for a day or so. Then disconnect it from the charger and let it sit overnight not connected to anything. Check the voltage in the morning it should be 12.6v or higher. If it's any lower I would replace it when the next camping season comes around.
Here's something else if you really want to dive into it.
Once you have a battery that is holding a fairly steady charge (ie 12.6 or 12.7v overnight, then take a voltage measurement on the battery terminals.
Write this number down and label it Battery Voltage.
Now check the terminals at the circuit breaker, thy should both equal battery voltage.
In the RV you will notice that the fuses have two little windows in them right next to the amperage rating. These windows are the opposite ends of the blades on the fuse. Poke the positive probe of your meter into one of the windows and find a good ground for the negative probe.

You should get battery voltage on each side of the fuses. If you read battery voltage on one window and 0 volts on the other window then the fuse is blown.