Hi Kelley,
I think you're on the right track with the scrap fabric for openers. I'd turn the iron a little higher and lay the scrap down with a parchment paper or teflon sheet and try a couple of times.
Then if there's some 'slime' that you can get at, scrap it off with anything that you can: spatula? popsicle stick?
Then: Do you have any Goop or Goo Gone? Or WD40? Or lighter fluid? I'd remove the ironing board cover and try using one of these products to remove the glue residue from the heat and bond. Then toss the whole thing into the washing machine and cross your fingers.
You might end up with discolouration on your ironing board cover, but better than replacing the whole thing.
Good luck!
PS Rubbing alcohol might also help.......