Based on what you said, I'm going to guess that I am one step above you on the tech-savvy scale. Here's what I do for Amex (and other) offers. The explanation is tailored to Amex; Chase is similar but usually has its own popup window and I don't need to print Selection Only.

After clicking to add the offer, a section of text opens with the terms and conditions. I select the text that is applicable to the offer I just added. Then I Print to PDF and choose Selection Only. When I say to print, I navigate to my "offers" folder, name the offer YYMMDD nnnn hint, where YYMMDD is the expiration date in that format, nnnn is the last four digits of the credit card number, and hint is where the offer is valid. For example, 220929 1234 Lands End is an offer for something (I'd need to look at the file to know what and how to get it) at Lands End that expires Sep 29 and is valid on my card ending in 1234.

It works for me, and I'd be happy to learn of a better way to keep track of offers.