Narrow tires and bad shocks will definitely feel unstable.
I've got 255/60s on the rear and 235/60s on the front of my pickup and it corners very well.
The Suburban had 9.5/16.5s all the way around and also cornered well.

Stock rims and tire sizes roll real nice but radials came along after these trucks were built and have done wonders to improve handling.
I have 215/85 16s for a different truck and they are on 7 inch wide rims compared to the old 4.5s and it makes a world of difference.

How wide are your rims and what size tires are you running

Ray