Crawlers wont have that nasty side to side sway if the suspension geometry is setup correctly, unless you mean the way the suspension actually works when its *is* done correctly?
When you hit the throttle the middle of the truck will bottom out under acceleration, the same way a normall MT does when it has rear springs that are too soft, or atleast thats what I envisage..
If you just had a solid rear axle, then the suspension would only go up and down in the middle, keeping the rear arms level and thus giving you the same suspension feel as dumper truck
I think what your trying to acheive is something akin to the way a Tamiya madbull's rear suspension works; it has a sollid rear axle, with a shock on either side copnnected to the upper chassis, but then on the bottom it has a swivel joint as such that allows the axle to rotate about its center, lifting only one wheel as that side drives over a bump.
Might be worth looking at, otherwise I'd say just go for the extended chassis with normal independant suspension- bigger wheels like Harold suggested is a good idea though, maybe mulchers or paddles perhaps?...