Could be the stabilizer, or it could be a combination of things.
Steering box output shaft play.
Steering box
Trac bar
Start your truck, roll down drivers window, standing by drivers door, reach in and slightly move steering wheel, look above tire and you should be able to see steering shaft going into steering box and you should be able to see pitman arm, when the input shaft is moving is the pitman arm moving? if not you have slop in steering box.
Mine was like that. I had two inches either way of play in the wheel before the wheels would move.
Took back to selling dealer, did the steering gear adjustment, helped a little, took back and had them do it again, little better. Then I moved and couldn't get back there "Big O Dodge" Greenville SC, they were good to me. I took it to two other dealers and both said thats the way its supposed to be.
Fixes I've tried:
DSS-Darin's Steering stabilizer- helped a lot but still sloppy,
Sky Jacker's D25 springs
Bilstein shocks to go with the springs- These helped the ride but didn't affect steering.
New steering gear- Ah finally, steering nice and tight, didn't know till now I needed a new trac bar, mine was so loose I couldn't tell.
DTPro Fab Trac bar- to go with the Springs, and replace POS stock bar.
Hook up with some TDR members and have them look at it, they will have ideas.