You cannot credit JetBlue flights to BAEC at all, and even AAdvantage lets you credit only certain routes (check here), so, short of opening a TrueBlue account and putting them in there (they let you credit flights back up to 90 days) these won't get you anything.

Seeing you're living in the US the US and fly mostly domestic, AA seems to be the logical choice. Although the BA program has its advantages (namely cheap redemptions on short AA domestic routes and the option of household accounts where you can pool miles for the family) if you want to attain status on BA they require four BA flights per year, which, unlike the same on AA, is enforced.

Those US miles will only transfer to AAdvantage when the FFPs merge. If your husband and child don't have AAdvantage accounts they surely will get one created when the program gets rolled over.