I guess it would be a bit far to drive to the mainland and work there. Could you live on the mainland with the baby and work there and see your husband on weekends? Are there other medically-related opportunities in the keys, such as urgent cares or free clinics? Keeping active in medicine, even on a very part-time basis, will be better professionally than dropping out altogether, although sometimes that can't be helped.
You should definitely keep your state license and your DEA license, if you have one. I don't know how the peds boards work, but could you study for them and take them if you are not working as a physician? If so, that might be a good thing to do with your time at home. And even if you are not studying for your boards, you can always do CME at home. There is a ton of CME available online. I found that it was pretty easy to read while my baby nursed.

There are also nonclinical things you can do (there are some threads on it here, if you do "forum search"), if you are interested. Or you could just use this time to strengthen your bond with your baby.
Whatever you decide to do, congratulations on completing your residency, and enjoy your new baby!