Not a painful needle involved time!
I emailed out minister and this is what he sent me:
This is a message i got from our wedding coordinator...
Whit, I called the Ouachita Parish Clerk of Court and the wedding license she gets here in Louisiana is ONLY good in Louisiana (any parish) -- this goes for Covenant and "regular" (!) marriages. You cannot use a Louisiana Marriage License in any other state.
Looks like there isn't going to be a covenant marriage for us...but we are lucky enough to get to have a Mississippi blood test whoo hoo(!).