Installing Dreamweaver CS4 to new computer, but the serial number is not accepted

I have searched the FAQ on this site, but can’t find anything that helps. So here’s my case:

Some years ago I had purchased Dreamweaver CS4 and installed it on my Mac. Now I got a new MacBook Pro and would like to install DWCS4 there. This turns out to be a major problem. I’ve already been through a lengthy chat with an Adobe representative that didn’t resolve the issue. She told me I should ask in the forum. So here’s the jist of my problem:

I can’t install DWCS4 from the CD I have, because it claims that the serial code, printed on the box, is incorrect. I don’t understand how that’s possible.

The chat person then sent me a link to the download site for the CS4 software that should work for me:

     https://helpx.adobe.com/creative-suite/kb/cs4-product-downloads.html

I was able to download DWCS4, but when trying to install it, my serial number again did not work.

That’s the same trouble I had when trying to install it from the CD I have: the serial number

which worked when I installed it on my old Mac is now not recognized.

The chat representative tells me I should go to the licensing website:

     licensing2.adobe.com

But no matter what I do, that site always complains “Adobe does not recognize you as an authorized user of this web site. Please contact Adobe for assistance.” But I have an Adobe ID and I can log in to my Adobe account just fine. Why does this site suddenly not like me? (I removed the history from the web browser, didn’t help.)

My license key is not registered with my account, either because I forgot back then, or because I don’t remember my account details. But I told the chat representative, and she tells me that she associated the license with my account. But it doesn’t show up in my account. And when I alternatively try to register it with my account, the system says that the number is already in use. (Of course, by me!)

So here we are: I have the product and the license number, but I can’t install it, because the system doesn’t recognize the number. But I also can’t go to the licensing site, because that site doesn’t even let me in. I don’t know what I’m doing wrong. Something seems messed up. I really would like to continue working on the webpages I have developed, but getting the new computer seems to cause a major hurdle here. What do I do?