Start with the first things first. An aftermarket stabilizer kit isn't going to stop the wobble. Steering stabilizers, which have been used in OEM applications for decades (there was a factory one on my '67 Bronco) exist to reduce steering wander on the highway. Basically some resistance to motion is being introduced in the form of a shock absorber. The real culprit is that the front end doesn't have enough positive caster to keep the wheels straight at speed. On early trucks without power steering they needed to run very little if any positive caster because the more caster the harder the wheels are to turn. With power steering this issues goes away, because you will not perceive a difference in steering effort with power assist.
On a 4WD you can't make big changes to caster, because you are typically limited to a max of about 1-1.5 degrees using an offset upper ball joint bushing. The real way to get good caster is to cut the knuckles free of the rear housing and reindex them with the ball joint centerline inclined more rearward as viewed from the end of the housing. That's a lot of work as those knuckles are completely welded on there and if you can't do it yourself you'll need to find a good shop that can handle it. The problem can get worse from lifting the truck. I'm assuming that you are running either dropped radius arm brackets, or longer radius arms. The problem is that if the drop brackets are less than the lift height you've lost caster right there. Longer arms can make the issue better, but can also carry the same drawback. The idea with longer arms is that you can lessen the angle of intersection with the frame and thus maintain caster angle, but there is a practical limit; the arms can't mount to the centerline of the rear axle for example, so longer arms may make the situation better, but may not be able to fully correct it even back to the stock geometry.
Compounding the issue is that bigger tires place higher loads on all front end parts. These trucks are notorious for wearing out ball joints (I'm on my third set). I would diagnose the suspension first and make sure that all the stock parts are up to snuff. I will bet that you find something worn out. With the front tires off the ground grab the wheels at 12 an 6 o'clock and try to move them towards you and away from you. If you get any movement the lower ball joints are shot. I'd recommend the NAPA or Moog lifetime ones because you'll never pay for the parts again, they are greaseable, and built better (all metal, no plastic).
Another test is with the tires on the ground have a friend rock the wheel back and forth (just sort of shake it side to side, you're not trying to move the tires) while you lie under the truck and look for anything moving. Track bar bushings and joints can wear out, as can tie rods, and drag link ends. If something's worn you'll see it. It will look like its dancing back and forth while everything else doesn't seem to move.
Give that stuff a try and it should fix your issue once you find the culprit.
'06 F250 4x4 - 5" Flo Pro exhaust, SB Filter intake, Accufab elbow, Edge Evolution (monitoring only), SCT w/ ID custom tune, FASS 195 pump, Gillette Diesel EGR cooler delete, Sinister Diesel coolant filter, ELC coolant, updated turbo drain tube/oil feed line/STC fitting/oil cooler, ARPs w/OEM HGs, Elite coolant lines, ITP RR fuel system, RCD 175/30 injectors, Powermax, BD CCV, FICM.com FICM w/ ID tune, Elite UP, BPD water pump - 13.069 @ 101.94
'02 WRX - Outback rear disc swap, EBC green pads, DBA pillar vane rotors, TXS UP/TBE/TMIC, Perrin LW crank pulley, PPG billet steel shift forks, ACT Streetlite flywheel & clutch, K&N filter, STi Group N motor/trans mounts, TiC/Kartboy rear diff mounts/subframe lock bolts/outrigger stiffeners, Kartboy SS & all shifter bushings, custom PDX tune for Cobb AP - went 14.1 on a terrible 60ft before most of these mods; shooting for 13.50s