The strut is what is keeping your wheel upright. When cornering, the side pressures increase and having the strut wobbling shouldn't be a big deal, unless it folds over and no longer controls the turn.
I would take the nut off the top of the strut, then jack it up at the A-arm, put a jack stand under frame, remove wheel, and lower the A-arm. When the strut comes free at the top, check it for resistance (it should be pressured up to the top), if no resistance it is shot.
You should check that your bump stops are long enough to keep the struts from bottoming out. Bottoming out, or topping out your struts are the quickest way of destroying them.