As someone who learned to drive on an '83 Suburban, I feel well-qualified to answer this question! Between my parents and myself, we've owned 3 Suburbans (an '83, '87, and '95) and seven half-ton Chevy trucks, so I have a great deal of experience with these vehicles.
I have a Hotchkis TVS system (F&R sways) on my '94 Chevy K1500 (1/2 ton, 4wd, regular cab short bed). It has far less body roll than my '97 328is with the factory sport package 
I'd make sure your steering components and bushings are in good shape first. Depending on your budget, I'd get either an upgraded rear sway bar (Addco or something similar), or F&R swaybars. I run Bilstein HDs in my truck, and I've been pleased with them. Tires on my truck are 9 year old Firestone Destination ATs, which are positively awful (not to be confused with the new Destination ATs which get rave ratings on TireRack.com), so I'm sure I could gain a lot there by putting on some better rubber.
If you could only change one thing, an upgraded rear sway bar would be my recommendation.
Osterkraut wrote: No no no. Lose the B and the N in the name. Switch the U and the A. Problem solved!
SUBURBAN SUURAN SAARUN?
What?
SUBURBAN - B + N= SUBURA Switch the U and the A= SUBARU
I think thats what he was getting at.
and to the above posted, 9 year old tires? The tire industry recommends replacing at 6 year intervals, tread or no tread.
Tires sound like a good starting point. Weird, though, my 84 1/2 ton truck (same stuff, basically, as a sub from the same era) had 70 series rubber and super narrow 195 tires (for snow - all year round) and didn't display any unacceptable traits. I would routinely slide the rear end around wet corners, drag race it (17.10's all day long) and cruise on the highway at normal speeds with no real fear or discomfort. All of the suggestions on here are fine, but I think there is something basically wrong with the truck. Did you buy it new? It just sounds like there is something inherent, because no newer vehicles are that bad.
andrave wrote: SUBURBAN - B + N= SUBURA Switch the U and the A= SUBARU I think thats what he was getting at. and to the above posted, 9 year old tires? The tire industry recommends replacing at 6 year intervals, tread or no tread.
Yeah, that took me a bit to figure out... had to realize first he meant different only one of each of the 'B's and 'U's in Suburban, then which ones he was talking about.
vazbmw wrote: ya, you have said a mouth full, but it has been floppy from the beginning. I think it has loosened up to the point of unacceptable to me now. That, and the fact my BMW is so tight and crisp.
That brought a smile to my face, I can't comment on it, but u sir are brave.
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