Fixed by:
1) Setting the panniers farther back on the rack,
2) Using a rack with more setback, or
3) Using a bike with longer chainstays and is intended for carrying panniers.

There's also option D, which is a set of panniers that can be set at an angle. I saw a sales rep at my LBS showing them off, but I forget what brand they were.

Point is, for every problem we encounter, somebody else with engineering and marketing savvy has come up with a solution and is selling it.