If you just bought it used, I'd start out small, and change the hydro fluid on both sides, and the filter. Then fill it back up with some quality stuff, jack up the rear end, run the handles forward and back to expel any air trapped in the system/bleed it so to speak.
See if that eliminates the wheel shake.
Do it after you have have mowed, like a regular engine oil change, might just be such old fluid it's got water, contaminates, whatever in it, and it will drain out. My .02c anyway.