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An inquiry won't likely impact your mortgage ability all that much. I have no idea what your scores are but, 1200 isn't really worth the hit if you're so close to buying a house. Just pay it off at an accelerate rate to avoid as much interest as possible but, save your apps for after buying the new place.
I guess the other thing to add is carrying very little CC debt going into it factors better for your DTI calculations since they calculate the CC min payments for each card that has a balance into the payment you can afford. So, if you have a balance on 6 cards w/ $35/mo min's you're shooting yourself in the foot for $210/mo less in home that you can purchase. $210/mo on a 30 year note is a ~20K in purchase power you lose.
Yup, pay off the interest balances first and then revert back to the 0% balances. Before applying for the mortgage pay them all down as far as possible to score the most points for your mortgage scores.