Sounds like you've got one of the pipes running to close to the chassis. When decellerating, the flex in the motor and tranny mounts is twisting the engine, picking it up on the passenger side (i think) and reducing the clearance on the RH pipes. I'd check there first.
'66 A-code Fastback (therapy)
'89 Town Car (SWMBO D'driver)
'87 Caprice (as req'd by Hagerty)