Quick update here... updated to 6.2.8 and experiencing the same issue, faster tho.
Watch reports memory full within 1-2 hours of completing update. Previously this would take days, later 3-4 hours.. now its faster.
I updated my feedback ticket with screenshots of the About screen, Usage screen, and Diag log screen. The watch is producing similar logs as it has over the course of this problem, a few less, but similar titles.
I was spending hours capturing, logging, and uploading this content since may with no response through the feedback ticket. No idea if it's going any where. Apple continues to point to updates as a resolution.
I suspect this has more to do with data in my iCloud account not syncing correctly, as the watch was fine with left unlinked to an Apple ID.
Thats it for now. Encourage everyone to update their OS, lots of security holes filled this round, and if you have time consider uploading logs from your devices through an Apple Feedback Assistant ticket.
This thread is getting more and more views, up to ~250 as of this writing... maybe a more wide spread issue.