If the univolt is bad, it could drag down a good battery. If you recently bought a new battery, disconnect shore power, disconnect the battery, and measure the voltage. If it is below 12.6 volts, put it on a charger for 12 hours or so. Disconnect the charger and let the battery rest for a couple of hours. Measure the voltage again. If it is not 12.6 or higher the battery is bad. Before you reconnect it in the trailer, put the battery cables so they will not touch each other or any metal part of the trailer. Plug in the shore power and measure the voltage at the battery cables. It should be over 12 volts and your lights should be at full brightness. If not, the univolt is bad.
If you have to replace the battery, get a deep cycle RV/Marine battery as a minimum. If you have to replace the converter, there are several sources, and maybe someone on here will chime in. I bought a Progressive Dynamics 9260. Others on here have recommended Best Converters as a source of help to decide which converter to buy.
"You cannot reason someone out of a position they have not been reasoned into"
Al, K5TAN and Missy, N4RGO
2002 Classic 30 Slideout
2001 Safari 25 RB Twin (Gone, but not forgotten)
2013 Dodge 2500 Laramie 4x4 Megacab Cummins
WBCCI 1322, TAC FL-39, AIR 82265