Thanks so much jrnymn7,
I'm truly a novice at this. I can probably look through the paperwork to get the specs on the converter. It's the original from 1989.
The coverter is having one know about. It converts the AC to DC when plugged into shore power, but it doesn't charge the battery. My dad talked me into plugging a battery charger/maintainer in a plug in next to the converter which is hooked into the wiring to charge the battery when plugged into shore power. This did solve that problem. I thought I should buy a new converter, but he assured me this would do the same thing for a lot less money.
I bought the deep cycle battery at Walmart, so I know it's not the best. I just had a couple of hours to look at it yesterday. I won' t have anytime to look at it tonight. I will be able to make some time this Saturday.
I don't think my dad owns the type of meter you're mentioning so I'll need to buy one of those. I'm going to copy and print your advice to reference when I have time this Saturday.
I was able to get a great price on the trailer. The interior is in good shape and there's no dry rot. I'm slowly getting everything figured out. I was having issues with the furnace until I pulled out the motor to discover the bearing grease was pretty much all dried up. I was able to figure out a way to take care of that so it spins freely once more. The fridge has been replaced and is practically new.
The person who owned it before me used it mostly with full hookups, but had used it when hunting also. I can get more specifics on the battery/ wiring/ converter this weekend.
I appreciate everyone's advice. I do want to be able to dry camp and I do need to make sure everything is taken care of before I make any plans for a multi-day trip without shore power.