Unfortunately, it is impossible to be 100% sure if anything is completely safe. It's not just the shrink wrap, it's the age too- certain chemicals react with air/sunlight, so your shrink wrap will have unique properties even other old wrap is not guaranteed to have.
Your greatest weapon is going to be: PAITENCE. You're not going to be able to get a good scrub of anything, so all your options will take a lot of time. This is not a 10 minute project. This may be a multi-day project. Don't rush.
I've used lemon furniture polish as an adhesive remover for years, so that's a secondary option for you if you become concerned the goo gone will be too harsh (but I doubt it, so don't worry on it too much.) If you do decide to try it, spray a good pile of foam on a rag (you need the foaming kind), then use a paintbrush to apply over your adhesive. Let it set a minute, then do the thing with the qtip.
I'd skip the wet tissue, personally- if the adhesive retains (or regains) any tackiness, you run the risk of getting tissue bits stuck to your game. Get a soft microfiber cloth (mine are from the dollar store & the bargain section in front of Target, they're cheap enough) & gently wipe away the residue. You should have enough residual moisture from the cleaner to help the cloth absorb the gunk. If not, have a second microfiber cloth you can get damp so you don't leave tissue bits behind. If you do have the furniture polish, you can use it instead of water to dampen the cloth.