If anyone says towing with full tanks is no problem, I say that they are playing with fire.
The first thing you should do if you want to do this is go weigh your trailer at a scale and see what the actual wet/loaded weight is and then compare that to your GVWR. Then, and only then, decide if you want to tow your trailer and exceed the GVWR.
Many trailers weigh too close to the GVWR when loaded for camping. If we were to travel with a tank full of fresh water, we would be overloaded. If we were to travel with all 4 tanks full, we'd be more than 1,000 lbs over the GVWR. I am not going to do it. If you really want to and your trailer is still under warranty, I would see if your dealer will give you something in writing that will cover a crack or failure if you tow with one or more full tanks. I doubt they will do it. The first thing Lippert will try and say is that you overloaded your frame and walk away from it.
For those that insist that trailers should be and are built to handle the weight of water, then you would do good to start reading all the posts you can find on the various RV forums out there about owners who have had cracked frames, failed welds and broken tank support straps from overloading their trailers. It happens a lot more than people realize. Frames these days are being built on the thin edge of barely being able to support just the UVW. Frames typically aren't even I-beams anymore, they're 3 pieces of sheet metal welded together to look like an I-beam and they flex like crazy. If you knew what your frame was really like, I doubt that anyone would think it was no problem.
