Where shocks come into play on straight road wandering problems is if the suspension geometry allows for bump steer. Bump steer happens when as a result of suspension movement, the steering links pull on the uprights and cause the tires to toe out or in excessively, creating a steering input.
If the shocks are bad or the vehicle is under damped or under sprung, and a bump steer geometry exists, then you'll obviously get it every time the coach porpoises.
You can bandaid this with stiffer springs and shocks and thus limit suspension movement or change the time constant so that it's more correctable at the steering wheel, but when it gets right down to it the best solution is to correct the geometry problem, usually by putting spacers between the tie rod end and the knuckle.
solid axle vehicles don't have bump steer, but they will have roll steer (which just means that to generate bump steer with a solid axle one side has to go up while the other goes down. You can see an animation here: https://en.wikipedia.org/wiki/Bump_steer
If this is play (or compliance) in the system, then one needs to get under the coach while it's running and have someone steer the vehicle while you put hands on the various pieces and look/feel for differential movement. Obviously this is dangerous from a "you might get run over" standpoint and from a "getting your hands cut off" standpoint. With the coach running you'll need something like an FRS radio to communicate with the driver because he won't be able to hear you give directions.
My neighbor and I did the method above to find out the culprit behind the steering vibration in his mini-van. turned out to be a bad rack, and the only way to tell was to have the tires on the ground, one hand around the shaft and another on the rack end, and then you could feel the play between the two. Was too subtle to feel with the wheels up in the air and no load.
Steering box slop ought to be obvious, but sometimes there are other things that come into play, like brackets that flex, etc. Don't overlook the pittman arm either.