I bought the full game and no it won't let me play and says that I'm in the demo
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I am not a dev but I would like to help. This usually happens when the user did not install the final build. The way to get the version you bought is to download the latest build from here on itch.io and just install on top of demo/existing build using sidequest.
If you click on the latest development log above (version 1.1, the one that says "Multiplayer open beta..."), there will be a link to a “manual” which I believe will detail exactly how to install the game. Good luck and let us know if you are still having issues, we will get you up and running!