I purchased some elephant garlic from Walmart after it was beginning to break apart on its own, the center was a spongy consistency, but there hasn't been a shoot from any of the bulbs after a week. Should I just dig it up, and use it, or wait for a few more weeks to see if it will sprout?
Did you purchase seed garlic that was spongy? If so, take it back and ask for a refund!
Garlic is supposed to be planted when there's a chance for frost. The cold stimulates bulb formation. Otherwise you will likely just end up with tiny bulbs and a lot of leaves.
And, if you're planting from garlic that was meant to be eaten, well, you take your chances for reasons that have been discussed many times already here.