A new battery will not necessarily have a full charge, but it should be decent unless you've had it sitting around for a month or more. I have not found driving to be a reliable way to charge the RV battery and I mean driving all day not just 20 min.
In the same way, the RV converter is not going to charge a battery in a couple of minutes. Don't forget the converter puts out 12V on its own so all 12V stuff will work even with a mostly dead battery as long as you're plugged into 110V shore power.
If your battery really is dead, I would charge it with an automotive type charger that has lights and a meter so you can tell what's going on. If you don't have access to one, plug into shore power for 3-4 days.
plugs into the 12V receptacle and is really handy for keeping an eye battery charge (I noticed the link wasn't working. Go to Campingworld.com and search for 19983). I leave mine plugged in all the time, but, like Steve said, this is one of the things that will slowly bleed off the battery charge. I have a battery cut-off and use it when we're not camping.