I'm not sure what's happening then if no error messages are showing up in the front-end when you've turned on the "Enable Debug Mode" setting in JFBConnect. There was one other user that had an issue with the migration where users that had authenticated previously had problems after moving to JFBConnect.
However, in that case, it showed a clear error when the attempt was made that helped narrow down the problem. The issue, in that case, was something to do with their LinkedIn application setting. We think it was that the app was created before November, 2012. There was a migration for LinkedIn Apps that occurred in June of last year, and if you didn't perform the migration manually (just checking a box in the LinkedIn developer area), the migration was done automatically. We think there was an issue if it was the automatic upgrade.
Ultimately, that user created a new LinkedIn Application and used the new ID on their site. That's not generally recommended as it a) makes users re-authenticated/approve your app and b) LinkedIn uses different IDs for users on each App ID, which means old connections in your User Map area won't be accurate until they re-authenticate and JFBConnect updates to the new ID returned by LinkedIn.
You can try to create a new App and see if authenticating again properly logs you in and updates your User Mapping to the new account properly (it should). That will help narrow down the cause and a new app may be an option for you, but I wanted you to know the above first.
I hope that helps. If the above doesn't sound like you, let me know. We can try to look at your JFBConnect settings to make sure things seem right as well if you'd like.
Thanks,
Alex