I actually just had a similar issue on my real NES/Everdrive. I used the Everdrive rom file with it. I just got a game over and went to continue from the title screen and it just started from the beginning again.
I wasn't super far in yet, but I suppose I will use save states in the meantime and see if it starts working at a later point in the game.
The game is very fun so far and I am enjoying the story!