Suburban Owners... towing/steering question ??

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Hello, I usually mess around in the 67-72 4x4 section but I also own a couple 88-98 trucks too... I recently bought a fairly decent 92 c1500 5.7 suburban to use when towing our small camper and my small car trailer when we have the need to also haul the family instead of just me and the wife. First trip out towing last weekend I noticed it needs some attention before regular towing use. It's almost like it gets a (secondary reaction)?? in corners at road speed. Like a terrible rear tire sidewall roll as the trailer weight pushes it in the opposite direction of the corner or something.. thus trying to turn the vehicle even sharper than intended. Honestly, it gives a little fist clenching reaction on the ole steering wheel as you try and correct.. just a weird feeling.. I've always towed with pickups, this is my first suburban. Never used sway bars or load levelers or any other components as my Camper is just a 16 foot and should be very manageable for the vehicle. At least always has been with my 1/2 ton pickups. Suburban Hitch is heavy duty and secure, It has 235/75/15 lt tires and all steering components are in good shape, Although front sway bar bushings are KILLED... Some of my pickups don't even have front sway bars and they tow fine, but again.. not suburbans I guess.. Will they make that much difference on the suburban? I've already bought new and will be installing as they are needed and I know they can only help all areas. But just wondering others thoughts ?? I played around with tire pressure already with no change... So higher ply tires, maybe front sway bar bushings will fix all, What am I missing here ??? Many Thanks in advance!!