Ron,
Thanks for the compliments on the new design. There's definitely a few adjustments to be made, including the "Hi Ron" (or whatever) text to let a user know who they are logged in as.. things we only noticed after live.
As for your question, yes, JFBConnect should work on any other site without a subscription. The Autotune tool will not be available to configure your Facebook application on the new site, so we'd recommend you do that while you're subscription is still active. The Open Graph Action features of JFBConnect do rely on some of the data that Autotune fetches from Facebook about your application, so without that, the Actions features will likely not work. However, Login, social widgets, page invites and just about anything else should work as expected.
I hope that helps,
Alex