We had a similar situation, but ours were good for 5 months. We called Jet Blue & they said it would be ok with St Martin, but would still be at the discretion of the airline, the day you are to go...
You may want to check into going in person to get an expedited passport.
Also, take a look at rushmypassport.com It seems you can do it at several Fed Ex locations in CT and get it in a week.

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