Since your subfloor is only 5/8", I recommend installing 1/2" underlayment grade instead of 3/8" The 1/2" will stiffen the floor a lot more than the 3/8". Underlayment should never be glued to the subfloor. If you used construction adhesive, it would actually creat voids and make the floor more spongy.
Good move to get rid of that stickin' carpet.:thumbsup:
Jaz