i got a thought, why not just swap out the whole front axel? or both axels, that would eliminate the axle completely, if the problem still exists then its either in the frame or the tires.
a shake like your describing would suggest something out of balance, but you have already addressed all of the possibilities i think.
some more thoughts i have, steering damper? is it new/good, engine mounts, are they good? cab mounts? i cant remember, have you looked at the control arm bushings? rear spring bushings? are the shackles and centering pins all ok on the rear? your problem seems to suggest something that is fundementaly wrong, like maybe the centering pins on the rear springs are worn out, or the axle can slide around some. this could cause alignment issues that are inttermittent, as it might be out of position one time and not another, i know you have had the frame and axles checked out by proffessional shops but i have never ever trusted anyone to inspect my truck properly, especially when there is a problem such as this with a difficult solution. reason being they dont want to think about it, or really try to solve it, they will do what you ask and when a normal inspection checks out they can cover their ***, look at you and say everything is normal and we dont know what the problem is.
what i would do is strip the axle down and look down the tubes from one end to the other, you will see if it is bent or not very easily, its how we found my cousins bent housing, if thats ok start looking at all the mount, rubber mounts can look ok and not be so test them, shake them, load them, stress them, see if there is any abnormal play, make sure all the bolts are tight and none are worn out. then i would be looking at centering pins and suspension bolts, same thing, bent, worn, or loose, also i see you live is iowa rust is a major factor up there and can hide a lot of problems, so clean the hell out of all the metal parts and then inspect for rust damage. something is moving in and out of position thats why the problem is so intermittent. i had a broken track bar bracket for a while on my truck and it would drive perfectly fine 99% of the time, but if i hit a bump just right, or the road was just the right roughness then it would death wobble. my last thought is how are your shocks, its a real stretch but at this point i wouldnt throw anything out as a possiblity.