K.E.
I know it can be nostalgic and kind of cool to use stuff from our childhood on our new babies, but anything that grosses you out and has things on it that you can't identify should be thrown out.
Grandma should get the biggest thank you and I would let her know how special it is that she saved things from DH's baby days. That does not mean you have any obligation whatsoever to use it.
Diaper rags are so inexpensive that you could just buy new ones and not spend your precious time cleaning/de-grossing/etc. these things. Good luck and congratulations on your new baby.