What stabilizer would you suggest?....The BFG's have been on truck for 4 years with no issues, have even wear and still good tread but I wonder a bit.
That is about 3 years too long for those tires. Same exact problem on a set of BFGs on an 01 and 2 05's, sent the 01 and 05 both into DW with just a leveling kit on them. Replace the tires and problem gone. Put them on my truck which was stock height with a dual stabilizer and good shocks, no DW but you could feel those tires bounce periodically on smooth roads. Just an intermittent bounce and if you moved over in the lane or changed speed it would dissipate then come back once everything stabilized. Even though they were an E rated tire with less than 20k when the problem started. The sidewalls are just not enough to handle the weight of the Cummins over time, they get soft and when they warm up they get magnitudes worse.
I like the dual stabilizers and even with worn out BJ's, tie rod ends and plenty of slack in the bushings ZERO DW. Bilstein shocks, a dual stabilizer, and a GOOD set of tires like Nitto, pretty much anything EXCEPT BFG, and DW is not an issue. If you have a steering stabilizer to tie the fornt frame rails together and support the pitman arm on the steering box get one first. Most of your problems are going to disappear with the tires, stabilizer, and SS.