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I have never heard of Adobe Export... do you mean you are using Acrobat?
You do not transfer programs, you do a fresh install
Download & Install instructions https://forums.adobe.com/thread/2003339
-and many help and download links https://helpx.adobe.com/download-install.html
-kglad links in reply #1 here https://forums.adobe.com/thread/2296468 may help
-you will need to enter your original serial number during install for non-Cloud programs
-Cloud Getting Started https://helpx.adobe.com/creative-cloud.html
-Membership questions and problems https://forums.adobe.com/thread/1146459
I see that you have subscribed for Adobe Export PDF service.
As mentioned above that you have changed the computer, so you can easily access your existing subscription without purchasing it again.
Export PDF is an online service, so either you can use it online on https://cloud.acrobat.com/ or else you can download the desktop application Adobe Reader DC Adobe Acrobat Reader DC Install for all versions and access through that.
You just need to "Sign In" with your Adobe ID under which you have registered Export PDF.
Let us know if there is any trouble in the activation.