Will my photos still be in Lightroom if I download my software on another computer?

(Note that you’re asking about Lightroom but this is posted in the Photoshop community, so it might have gotten a better answer faster if it had originally been posted in the Lightroom community.)

Creative Cloud lets you be signed into two computers at the same time, so that makes it easy to have both a desktop and laptop, or both an old and new computer which is what you’re doing.

When you get the new computer, download the Creative Cloud desktop app and sign into it with your Creative Cloud account.

Your screen shot shows that your plan includes Lightroom. Lightroom always stores originals on Lightroom cloud servers, not on the computer. That means you always have access to all your photos in the cloud regardless of which form of Lightroom you sign into: The desktop version on macOS or Windows, the mobile version on iOS or Android, or the web version that runs in a web browser. So when you sign into the Creative Cloud desktop app on your new computer and install Lightroom, all your photos from the cloud should appear in Lightroom and you should be good to go.

After you confirm that you can see all your photos in Lightroom on the new computer, when you are ready to stop using the old computer, sign out of the Creative Cloud desktop app on the old computer, and it will no longer have access to any of your Creative Cloud files. Then the old computer will be ready to sell or recycle, as far as Adobe software is concerned.

(If you used Lightroom Classic then it would be a different story because that stores originals on the computer, not in the cloud, but Lightroom Classic is not listed on your plan. So you don’t have to be concerned about that.)