Empty trailers bounce all over the place, and will really amplify tire imbalance. Try letting air out of the tires, down to 18-20 lb and see if it still shakes. I've hauled new trailers and have reduced air pressure down to about 12 lb just to keep them from shaking and bouncing when transporting for dealers. Do you still feel it with a good load on it?
If I moved 30 miles, I could join the Alabama Gang group