Use the Apple menu and select System Preferences.

Select the Printer preference pane by clicking the printer icon

Select the Follow Me printer.

To the right of your print job you should see a circular arrow. (See image below)

Click on the circular arrow and it will open a login dialogue box. Enter your new computer un and pw.  Your print job and future print jobs should then be released to the Follow Me printer queue.

If you continue to allow your computer to remember your user credentials for printing, this process will need to be done each time you change your computer password.