Rear tires are rubbing on fender!

I thought sway bars kept the cars body from leaning over too much.
Yeah but it won't add any clearance. You do need to stiffen up the rear end, and at the same time correct the ride height (up). Most easily done with new springs. Also, it looks like the wheels should have more backspacing (which of course the only way to correct is with new wheels). Or use a smaller tire.