Log out, clear your cache & cookies, close your browser and then start over. (This is in case you have something 'stuck' somewhere).
Then on the Annual Dues Overview page (the first link you posted), as
@kes601 said, there should be a button in the first box that says 'Pay by Jan 15'.
Click on that, and if using a git card, select 'pay by contract' and then, 'pay other amount' & enter the amount you want to pay. Then select the new card and gift card options and enter the card's account number and EAN.
Probably way more than you need to know, but maybe the details will help someone else.
I just did this with the COSTCO gift cards for our dues. It's working again, at least for me. Good luck!