Yes, GTUs had larger, stiffer front sway-bars, and in '88 ONLY (not ours -I see yours is the same year as mine) had a "Z51" code suspension which included a big external bolt-on rear sway-bar, which can be piggy-backed underneath the '89's rear sway-bar (mine has both)
. Lower, stiffer springs in the "FE3" suspension-code were only fitted to GT's as an option and GTU's, later Z-26s got them too... I think the FE7 "Active" suspension used on the Indys, GTZ's and one weird-code option GT, I think in '92, used similar springs, but MOST GTZ's were four-cylinder H.O. cars, and thus had lighter springs...
As for feeling squishy, fresh bushings and end-links for the sway-bars and control-arms might help, I guess if the struts are worn there isn't much you can do now, and if the springs are tired you are definitely facing an expensive uphill battle to refresh them to as good as new... I am in the minority for disliking Poly bushings -they squeak, crack, and slam old metal parts together too hard for my taste, but they do stiffen things up a lot.
A strut-tower brace will work wonders for the ride (my GTU, one of my Z-26s, and one of my Indys have them already, I think all of them will eventually have them), and those are still around, you can use late-80's J-Body (Cavalier) strut-bars too. Mine are all hacked-in "H"-Body braces, but I have reasons for using that particular tower bar. Rear shock-tower braces are out there too, I never got around to making some for my Berettas, but I don't think they need it as badly as the front of the cars do. Mine are all pretty solid still too -rust hasn't been a problem on any of them luckily.