If you're in Wayne county and it'll be difficult to make it to Downtown Detroit to apply for your marriage license and then have to go back to pick it up there is a Wayne Co. satellite office on Henry Ruff Road in Westland where you can go and apply for it and for an additional 10 bucks yiou can get it right then. We did that today. here is the address. I hope it helps you too!
Wayne County Satellite Office
3100 Henry Ruff Road