It's not a bad idea to replace your steering stabilizer, but that won't cure the problem. It comes down to balance and tire roundness, with maybe a bent rim or thrown belt somewhere in the mix.
Local fella has a tire shop; decent guy, he's even hunted down at our camp. But his tire balancer couldn't balance a cup on a saucer. Don't know if he didn't service the equipment or what. Such equipment gets used hard and is subject to wear. Also had some tires that were out of round from new, and they can't be balanced, IMHO. Not familiar with the road force machine, nor if it can solve that kind of problem.
Had a Honda several years ago that resisted all efforts to balance the tires. Finally pitched an extra large fit (hate to do that) and the dealer replaced them under warranty. Back during the Carteresque 55 limit, you couldn't get anything to balance; always guessed the tiremakers let QC go to hell because of the low speeds.
Look around for a tire shop with high standards and a good rep; try a balance there. Only way I solved a shimmy in my last XJ.
'04 Rubi, manual, usual options, Hella H-4s, any range lockers, Randy's booster seats, Oly pushbar, 245/75R16 BFG
'09 Audi A4 Avant- my bride's schnellpuffenpanzer
'12 Ford Focus-the sure way to get good Jeep mileage
'07 Dodge Caliber R/T AWD-da kid's ride
Goodbye Old Blue XJ; 10+yrs, 100K+ miles; virtually trouble free; Long may you run!