I wonder if the old stuff would clean off easily with
"Goof Off"? Be sure to try it on an unimportant part first.
Here from something I found on line and cut and pasted;
Celina Citrus Cleaner is a 100% biodegradable solvent that quickly removes adhesive residue from tents, inflatable bouncers, PVC, and plastics. This cleaner completely removes tough substances like engine grease, tar, paints, inks, adhesives, shoe polish, crayon, kitchen grease, and more. A diluted solution of 1:5 bleach to water can be used when general maintenance cleaning for more stubborn soils, but will not remove stains. Isopropyl alcohol or mineral spirits are also helpful in removing gummy residue or wet paint. Be sure to remove any excess cleaning solutions with a clean, dry cloth.