If you play on the Gamejolt page while logged in via the Gamejolt site, it should automatically log you into the game (you shouldn't need to use the login button in the game at all). Does doing it that way work? I'm not sure what else I can do to troubleshoot login problems, since almost all of that is handled by Gamejolt's own API :\
I have the same problem using both itch.io AND Gamejolt. Both don't like my Gamejolt game token. Whilst logged into Gamejolt and playing from Gamejolt, progressed isn't saved. Tried using the login feature (I am presented with the login option and New Game option - even if logged into Gamejolt. I even see greyed text "logged into Gamejolt..." in the lower right of the game screen whilst login screen is open. Not sure that made sense. :-) I suspect it's an API issue.
Keep in mind the GameJolt login has nothing to do with your saved game - the only thing logging in does is allow you to submit your scores to the leaderboards and earn trophies. Your saved progress is stored locally on your computer, even if you are not logged in. Troubles logging in may be GameJolt API issues, but you should still be able to save even if not logged in. Your progress is local, so if you play in the browser, the Continue button will only appear when you are playing in the browser on that same computer.