According to Wikipedia the non-convertible V6 has a curb weight of 3350 lbs, and the non-convertible GT is 3480 lbs. So only 130 pound difference. And that 130 lbs isn't all engine.
The heaviness you're feeling is due to the stiffer and tighter suspension of the GT, not the weight of the car. Stock GT's are not track cars, but they actually handle fairly well from the factory IMO. Give yourself a chance to get used to it before you start modding the suspension.
2008 Triple black Convertible GT Premium, Auto
Charcoal premium wheels, Appearance and comfort packages
Roush CAI, Bama Tune, FRPP axle backs.
Steeda strut brace, Honey comb rear deck lid, Sequential tail lights, Tail light frames, MMD side scoops, MMD styling bar.