We can move data from a free account into a paid account, or between paid accounts, upon request. Here are the steps:

If the same email exists and is an admin in both accounts:
Simply email us at support@ from the admin's address with the subject "Move Projects from [a1] to [a2]", where [a1] is the old account and [a2] is the new account. In the mail, include the projects you wish to relocate. We'll move them and notify you when the move is complete. 

If you have different emails for the accounts, or if you are not an admin on both accounts:
Have an administrator from the source account mail us with the request, and then record the ticket number. Give that ticket number to the administrator from the destination account, and have them mail us to confirm that they wish the projects to be added. We'll verify, then move them, and notify you when the move is complete.

Either way:
Other than your user, you'll need to add the other users you wish to have on the project in the new location. You won't lose any historical data if users who checked in to the old project aren't in the new account.