Roger, thank you very much for the images and the advice about making a wooden deck of cards to offset the side to side racking.
And, as both you and Jim suggest, I'm going to double-check the parallel rods to see that they are happily in alignment and tight.
I believe my vise must be a 53ED since it is definitely quick release and it came with a steel dog (which always seemed to me a metalworking aid not woodworking).
I've shimmed the moving jaw and the vise is working much better as of now. But I'm going to review all its parts this weekend and see how it goes.
Thanks again, everyone, for your help.