Can't add PS CS5 to a new computer.

Legacy CS5 was built to run on WinXP, Vista, Win7 or macOS X (10.5.7) all of which are dead now. There's no guarantee that unsupported software will work on modern Windows. And there's zero chance of it working on a modern macOS.

MODERN OPTIONS:
==============
FREE Acrobat Reader
https://www.adobe.com/acrobat/pdf-reader.html

FREE Online PDF Editor
https://www.adobe.com/acrobat/online/pdf-editor.html\

Paid Acrobat (Standard, Pro or Studio)
https://www.adobe.com/acrobat/pricing/compare-versions.html

Photoshop Elements or Premiere Elements (3 year license, no subscription needed).
https://www.adobe.com/products/photoshop-elements.html

Creative Cloud Annual Photography plan. 
https://www.adobe.com/creativecloud/photography.html

FREE or Paid Adobe Express
https://helpx.adobe.com/express/using/express-overview.html

Creative Cloud Bundles & Single Apps for mobile, web & desktops
https://www.adobe.com/creativecloud/plans.html

See system requirements. Most Adobe apps work on these systems, no more than 2 versions back:

• Windows 11 (64-bit) versions 24H2, 23H2; Windows 10 versions 22H2, 21H2;
• MacOS 15 (Sequoia), 14.7 (Sonoma), 13.6.7 (Ventura).

- https://helpx.adobe.com/creative-cloud/system-requirements.html
NOTE: MS is ending Win10 in a few weeks. Apple will release Tahoe later this year.

Hope that helps. 

you can try to reactivate on that old computer. if that fails and there's no other available computer with an activation, you're done.  you're limited to the 30 day trials with your cs5. there's no easy work-around.

if reactivation is successful on that old computer, you have a few substantial hurdles to overcome.  keep us updated so we can go through it.

@mmccarrell was your old computer connected to a network before you deactivated?