is it at all possible to create a free 'public' version of the game which has a link in it that takes you to the paid version of the game? I mean if that works then that would be a good way to still get traffic while having a way to also get money.
Yes, it's possible on Itch. In fact, all my games have links to my Patreon in them. Though, not being aware of these is a good indication on how effective they are :)