Thanks for the replies. I think things are OK now, but with a few oddities that I believe are probably inconsequential.
I suddenly got very busy so I'll post more later. But I will mention a few things now:
-- I have had Norton Security for years, always inputting a new Product Key the day the old one expires. I don't recall ever having a problem before, aside from the fact that retail NS keys (for the US, anyway) have become increasingly hard to find. I have never had a subscription or provided a credit card -- and as it turns out, that's still the case.
-- So this year I bought (what turned out to be) a 360 Product Key. (When I bought it, the seller said it was Norton Security Deluxe, but it turns out that per support it was actually for Norton 360 Deluxe. Read on.) After I entered the new "Norton Security" key, it said it had to take me to My Account to provide additional information (or something like that), but at the Account website it didn't actually ask for any information. So I searched Support at My Account on how to enter the product key (or something like that), and it just took me back to what I had been doing in the first place. So I was going in circles. (Somewhere in this process it was also offering that, with a subscription, I could "renew" Norton Security at a highly discounted price for the first year.)
-- So I called Support at the number listed in My Account. They were able to activate the key, although it took awhile to populate. While My Account now showed my Norton Security as expired, it didn't show how to activate my devices with the new 360 activated key. The Norton User Interface kept quickly switching to a "Product Expired, Renew Now" (or something like that) screen. Finally I found that if I clicked "Remind me later" it would show the old interface long enough to enter the product key. Then the Renew nag stopped, Subscription Status said 366 days remaining, and there was no longer a red warning that I needed to subscribe.
-- I then downloaded Norton 360 and installed it on top of Norton Security (i.e., I did NOT uninstall NS first). I then entered the Product Key, with no difficulty, into my Norton Security Installation on a second pc. And again installed 360 "on top."
-- I'll mention one of the probably-inconsequential oddities now, which is that on the first pc the top border of the UI -- as well as the desktop icon -- still say "Norton Security," even though the tooltip seen when hovering over the tray icon says "360." And in the program list, it is called Norton 360 (though in a folder that says Norton Security). On the second pc, ALL references are to 360. The exception being that again the app-list folder is named "Norton Security" (is that the normal app-list folder name with 360?). Regardless, it is clear on both machines that the actual product has changed from Norton Security to 360, given newly-appearing features such as VPN and Backup.
-- I ran the diagnostic tool (Help -> get Support) on the first pc, thinking it might change all the "Norton Security" nomenclature to 360 (even though, again, I doubt that's of any consequence). But instead, in two tries, it came up with the following "Attention" screen about its failure to fix Internet Connectivity problems...which is strange because I'm not having any Internet Connectivity problems. Nor is 360 as far as I can tell. For example a manual Live Update runs fine. I'll check later to see if any background connectivity-requiring tasks like Automatic Live Update have been running. And what the status of the browser extensions is, eg Norton Safe Web.
