Hi Haynumber05,
I am not with Laminar Research; just a flight simmer.
There is no password key for a disc version whether it be XP10 or XP11. You are basically stuck with what you have. Disc 1 has to be in the drive at all times.
Does your old computer still work? If so make sure XP10 is totally installed and located on the desktop or a dedicated drive of the old PC. The purpose is to ensure you have a complete installation and x-plane files not mixed into other directories.
Once installed acquire a decent portable USB connection style hard drive or portable drive with a "good" storage capacity (in excess of the directory capacity of your newly installed software on your old PC) and then copy the X-Plane 10 directory onto the portable drive.
Then insert this storage device into a USB connection on your new PC and copy/paste to your new machine. If done correctly and be selective of the download location then you should have a complete version of XP10 on your machine.
The best thing about XP10 and XP11 is the software can be located in a discreet standalone directory. If done properly it does not hide files in other directories on the hard drive like other flight sim packages.
Once completed follow the next part of my advice.
However you are in luck with version 10. Laminar Research created a USB dongle (I have one) that substitutes for the use of XP10 - Disc 1. It works fine. No problems.
You need to make direct contact with Laminar Research to ascertain if it is still available and make arrangements for the purchase. Details of the unit can be found at https://www.x-plane.com/product/x-plane-10-home-use-usb-key/
Good luck
Glenn