So I'm sorry to say but it looks like it is not possible to fix at this time. All though deprecated, the issue was not with System.Jason.dll.
Digging into it I believe that Itch.io's API endpoint has changed, and the key generated only has access to profile info, not game info. I will follow up with Itch and investigate further.
That's what this plugin uses. The token generated during authentication is used to make the API call https://itch.io/api/1/KEY/me. Once that returns a UserID, another call is made to verify the purchase https://itch.io/api/1/KEY/game/GAME_ID/download_keys.
It seems the last API call mentioned has been updated.
I have made a post on the support forum, this is the link if you would like to follow also. Once I have an update, I can update the plugin.
https://itch.io/t/1114371/itchio-api-endpoint-change