Traps inevitably dry out. But not all traps are siphoned out or blown out while going down the road. Aerodynamics are more complex than one might think. I suggest you check your roof vents to make sure their caps are in place properly and not partially blocked or otherwise obstructed.
It's possible for the grey tank roof vent to allow passing air to create a suction or positive pressure in the grey tank that then sucks or pushes water out of the traps. While plugging the sink drains will minimize this, the "instavents" under sink cabinets can still allow water to be sucked out of the traps.
This is a pretty obscure thing to consider, but you might be surprised to see a missing cap...or otherwise dislodged vent cap on your roof. Also consider that unpredictable aerodynamic influences from other items on your roof might be contributing to the problem. If this "dry trap" phenomenon has been happening since you got the rig, the rooftop layout of all the bits and pieces above the roof might contribute to the problem.
There are several different designs for these caps that you might consider if yours is intact and "should" be working properly but isn't.
Freakish weather conditions with high winds blowing across the roof vents in homes can cause this to happen, so dragging your plumbing down the road at 65 MPH presents quite a challenge for drain plumbing.