I got the shakes, man!

I had a similar problem years ago on another truck I owned. I put the truck on jack stands and put in gear and took it up to 70 mph. Turned out one drive wheel wasn't concentric on the hub.
Another thing you should do if you haven't already done so is check the driveline angles. If the angles are a bit borderline to start with, even with airbags a trailer can push down and that half degree out gives you driveline vibration. I've had that problem myself also.

 

Get yourself a spicer Anglemaster, max angles on UV is 3 degrees, loaded or unloaded.

 

Shakes, shimmies and vibrations. They all have a different feel to them and by what you feel can help pin point what it is. Vibrations are more from higher speed components like the engine, trans and driveshafts. Vibrations are more like a hum or buzz feel. Shakes and shimmies are more from slower moving components like tires and wheel assemblies. Since wheels spin about 3-4 times slower than the drive shaft, depending on your drive ratio, they tend to develop more of a shake. Coincidentally, highway speeds is what generates a resonance frequency in unbalanced or out of round tires/wheels to cause this shake. A tire or wheel shake frequency is slow enough that you can almost count the pulses. Generally, if you feel it in the seat or floor, it is a drive tire. If the steering wheel shimmies back and forth, it is a steer tire. In your situation, I think you have a drive tire out of round. Bobtailing allows the wheel to bounce and your suspension and shocks are able to dampen it. With a slight load, the extra weight forces the bouncing to be more pronounced, but with heavy loads the trailer is basically forcing those wheels to run smooth as the weight over powers and totally dampens the shake and this is why you only feel it with a light load.

 

Out of balance,tire and or wheel.

 

It sounds like you might have a mis-mounted rear tire? We had new steers put on where one was off enough to feel a front end shimmy, took the truck back and they remounted it and said the last guy didn't use enough lube when the tire was mounted, so when the bead was seated it was off set on the rim. It cured the shimmy for us.
It makes sense a loaded trailer shakes less than an empty one, the weight or mass of the load is acting like a dampener.
Keep us posted on what the problem was when you figure it out!

 

Sometimes you do what you gotta do. I would get them balanced and see what happens. I was so picky about balanced wheels/tires that I started mounting my own and putting dynamic balance media in them because it's expensive paying someone to do it. I dread doing it, I hate it actually, and it's hard work when you don't do it every day. But it only costs me some sweat and time. I wonder if maybe you have a brake drum out of balance? I dunno just throwing the easy stuff out there. I had issues like this years ago with my truck and I know how it sucks to drive one like that but eventually I got it riding smooth as silk.