If you are already using Photoshop CC then all you need to do is go to the desktop application manager and install Lightroom CC. Both programs are included in the $9.99/month photography plan. Lightroom CC will be installed in addition to your previous installation. When you run Lightroom CC for the first time it will prompt you to upgrade your catalog. You will want to do that so that you can simply continue working with Lightroom CC. Take note that the icon for the new version of Lightroom is identical to the old one. And both programs will continue to function on your computer. After you install Lightroom CC and find that everything is working you can uninstall your previous version if you wish to.
I think it's also important to point out to you that Lightroom CC and Lightroom 6 are exactly the same program. But Lightroom 6 must be purchased either new or as an upgrade.
I don't full apprehend, have you paid an annual payment for the Photography Plan, and are trying switch to a monthly plan. If so the subscription terms, link is below
Alternatively you can upgrade from LR5, to LR6 using the link below (clicking buy, will take you to the upgrade page). Given that you already have Lightroom CC, included in your Photography Plan, this seems pointless unless you have more than 2 computers.
One other thought. If Lightroom CC isn't showing as an install on your desktop manager it may be necessary for you to sign out and sign back in. A lot of people, including myself, have had to do that.
Very many thanks for both of your mails. I'll give it a try!