The 110v outlets work only when you are plugged in to shore power. The lights are all 12 volt
. You should hear a definite hum from the Univolt if plugged in to shore power. It has gone bad if it is silent.
Experience with my '74 Argosy and its Univolt was that 12 volt
lights would work without a battery in-line but it's not supposed to be a good idea. It is possible you could have some lights function with or without a battery; systems did change over the years so this isn't as easy to pin down as it might seem. You will need a deep cycle battery easily obtained at a farm/fleet store (cheaper source probably) or RV or marine dealer. Having a battery installed is a prereq for the nationally mandated breakaway brakes, so you'll need to get one anyway.
Old Univolts are on the borderline of being hateful and cantankerous at best. They were good in their time but can run a battery dry if plugged into shore power as they were designed to do. Go figger... They should be replaced at the first sign of failing to operate correctly. See BestConverter - Inteli-Power Converters, Inverters, Solar Chargers
and research more at the Forums to help make a decision if you have to replace.