I've never used a stabilizer on a model A but its nothing new . I.ve found rusty old model A cars and front ends with rusted pieces of stabilizers still attached. I know that the seventies model CJ 5 jeeps that I had used them and they steered like power steerting when they were moveing, even with P 78 mudder tires.
The loose steering arm, alone could cause the shake, ofter referred to as shimmy. Bad toe in can also contribute. I measure the toe in at the height just under the wish bone. Then push the car backwards until the measureing tool rises to the same level and measure again. I set the toe in at 1/16 closer at the front. I can't balance model A wheels perfectly with lead weights. dyna beads poured into the tube may work better, don't know.
I can feel a slight embalance on the 19 inch wheels on my speedster at 65 mph, it smothes out at at 70.
