I'm not familiar with your trailer, so can't comment specifically about the way yours works. Two thoughts come to mind since you say you're new to this.
1) Most wet-cell and AGM batteries usually have a 5-7 year life span before they start showing their age and you should start thinking about replacing them. Depending on the original sale date of your trailer and whether the batteries have ever been replaced, your trailer could be approaching 5 years old and your batteries could be at the end of their effective life, especially if they've been treated poorly.
2) The lights shouldn't be *totally* dead immediately after turning off your generator unless the generator isn't actually charging them in the first place or there is no 12v power from the batteries going into your converter. If your other 120v AC appliances and outlets work when the generator is on and the 12v lights and radio are on when your generator is running, that suggests that your transfer switch is working at least to power your converter, which in-turn powers your 12v DC lights and radio. It could be something as simple as a dead fuse or tripped circuit breaker in the circuit between your converter and battery.
Here's a good read on everything you ever needed to know about RV batteries:
http://www.marxrv.com/12volt/12volt.htm
EDIT: I originally included this, but I think you know this already after re-reading your post.
3) If you're expecting the batteries to power the 120v AC circuits in your trailer, things typically don't work that way. The batteries only power the 12v DC equipment (usually lights, furnace, radio, slide out motor, water pump, fridge controls, etc..) and the converter converts the 120v AC to 12v DC to power the above equipment and charge your batteries, but it doesn't work the other way around. If your unit has an inverter, that will allow you to power 120v AC equipment (e.g., TV, microwave, electrical outlets) from the 12v DC batteries when you're not plugged in or running the generator. Pretty much every trailer has a converter, but inverters are much less common.