How to prevent iTunes from forgetting albums when switching Devices?

I had to purchase a new laptop (ASUS) to replace my old one (Toshiba) for school. I have found that I am missing entire albums of music that I have purchased through iTunes. When I got to my account, they were gone, so I could not restore them. However, when I booted up my old laptop and opened iTunes, the missing albums were still there. WTH is happening?

I will add that more than a year ago, I tried Apple Music, but I got rid of it when it kept replacing song versions. For example, a rock song was replaced by an acoustic 'live' version I didn't ask for, nor was it on the original download I paid for. In getting rid of Apple Music and many hours of work, I was able to restore what I could find (I have thousands of songs). I want to resolve my new issue without reactivating Apple Music, which will undo all the work I put into fixing it the first time.

I am well beyond frustrated, and I do not have hours to sit and do this again. Does anyone have any idea why my iTunes is "forgetting" entire albums when all I did was change laptops? The new one is authorized, so that is not the issue.

[Re-Titled by Moderator]