<quote=swizzle>Do your pots have drain holes? If not, try not to give your figs to much water. </quote>
I'd say: if they don't, then give them drain holes.
Welcome Jennifer.
To the good advice about varieties that you've gotten above from others, I'd add a few thoughts also. I live in a similar climate to yours (I'm in central NY state, USDA zone 5a). The best producers here (in terms of producing figs with good taste, and which ripen nicely, rather than necessarily in terms of numbers) have so far included Hardy Chicago (aka Bensonhurst Purple, aka Mangibello) and other very similar figs such as Sal's EL and Sicilian Red. (All of those are dark figs, and probably my favorites of the ones that have done well here thus far). English Brown Turkey has also done well here (breba crop only, not the main crop). Battaglia Green produced some excellent figs here last year (it's a green fig with a jammy red center). (I believe it is similar to Ischia Green, mentioned by Pete / ascpete up above). And for a somewhat different taste, Verte produced figs that had interesting crunch and a tangy / tart taste. I'm doing trials on quite a few others.
There are others that have reputations for doing well in a cold climate, including Nordland, Florea (mentioned by Ottawan and others above), and Tenica. But I'm still trialing those, and they're too young to have produced anything yet. I have a third year Florea but it hasn't yet produced any figs.
Good luck and welcome to the forum.
Mike central NY state, zone 5a