Do demo saves carry over?
No. Demo saves are not compatible with the full game. The demo and full game use different levels, so expect to start a new adventure in the full release.
Why did an older demo save stop working?
The September 12 demo update also made saves from the previous demo version incompatible. This is a separate issue from transferring a demo save to the full game.
Should I keep playing the demo?
Play it to try exploration, combat and party building. Treat its progress as a separate adventure, rather than a head start in the full release.