Sounds like play in the tie rod ends or steering box. The original style tie rod ends have a plug in the end that can be adlustec for play, you will need a drag link socket for this. You can get that from Snap-On. To adjust, remove the cotter pin from the end, turn the plug in until play is removed, install new cotter pin and re-lube. You will have to reset the toe afterwards as it will toe the wheels out. The steering box may also need adjustment, the adjuster is on the engine side if the box. Loosen the jam nut and turn the screw in to remove play.
Asway bar only comes into play on turns, what you are thinking of is a steering stabilizer. There is a universal one availible from Moog. Get part #SSD 12 Trail Boss. This atteches to the axle and the tie rod and is essentially a shock absorber. I put one on my '55 and reccoment it if you are not running power steering.
