For a plush that old I’d highly recommend hand washing in a sink or the like. The chemicals used for dry cleaning can damage the fur, so I wouldn’t suggest that. I’d love to see a pic of piggo if you’re okay with sharing, I love seeing old plush! I still have my little dog that was given to me at birth (44 years ago) and I recently had to do some cleaning on him, I honestly used a rag and dish soap to wash him up and it worked a mint without damaging him, something like that would let you avoid the markings. Just don’t use too much soap or it’ll smell like soap forever.
If you know how to sew it might be worth opening him up and switching out his stuffing for fresh, you can buy several different types at Walmart depending on which you like more. Ladder stitches are ideal for closing up this kind of repair, there’s infinity guides to do this online.
Sorry, you kinda activated my autism about stuffed animals haha