Hello timmo,
You don't need to move your license, you only need to input it in the new computer. Check the license:
Licenses are per-user, rather than per-machine, so you can enjoy Sublime Text on as many computers and operating systems as you wish with your license.
More...
If you have access to the email address you used during payment then you should have access to the key, do you not have access to that email?
- Eduan