I have a '63 LeSabre, and I don't recall that it has ever "bottomed out" while driving. I'm still running stock configuration exhaust on it, though. It has 225/75/15 tires on it, which are pretty much equivalent to the H78-15 tires of days gone by. It is possible that whoever installed the dual exhaust on your car bent the pipes in a way that allow them to hang lower than they should, and as you mentioned, a lower profile tire would also cause this.
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