Is there a way to keep my tires from scrubbing and busting?

I think you better dig into this a little deeper, the offset could be wrong, suspension components could be worn enough to cause camber and caster problems as well as soft suspension springs, after all it is a really old European import and things wear out.

My first guess is that the larger rims and tires are TOO BIG for the car. They're rubbing where the proper size 16" don't. Look at geometrics and listen when driving. If you can strap on a video camera to watch a tire, do that.

I guess a large hammer or some hands on fender work is out of the question ?