Matchstick
Adventurer
I did the Dungeons & Dragons Fantasy Adventure board game with my daughters at that age (6 and 7) and we had a great time. It has figures, premade adventures, and you can carry advancement to the next adventure. They kept their character cards in their rooms for a good long time because they were so excited. They role played to their comfort level, which was fine IMO. I certainly wasn't going to force them to di it!
Unfortunately, I'm not sure that game is highly available, or available at all. But I do think a RPG board game could be a nice transition between board games and RPG's for your family like was for mine. Maybe the Pathfinder Board Game? It's got math, carries over characters, plenty of adventures, etc. It might be a little complex for the 6 year old, but it also depends on the 6 year old.
You could look in to "Secrets of Cats" for FATE (needs FATE core, but both are pay what you want). I'm not a FATE guy, but I know dang well my daughters would love to play something with cats. Which reminds me, there's a very simple Cat RPG that's out for the Warriors series of books. You could take a look at that: http://www.warriorcats.com/games-and-extras/games/adventure-game
Hope that helps!

Unfortunately, I'm not sure that game is highly available, or available at all. But I do think a RPG board game could be a nice transition between board games and RPG's for your family like was for mine. Maybe the Pathfinder Board Game? It's got math, carries over characters, plenty of adventures, etc. It might be a little complex for the 6 year old, but it also depends on the 6 year old.
You could look in to "Secrets of Cats" for FATE (needs FATE core, but both are pay what you want). I'm not a FATE guy, but I know dang well my daughters would love to play something with cats. Which reminds me, there's a very simple Cat RPG that's out for the Warriors series of books. You could take a look at that: http://www.warriorcats.com/games-and-extras/games/adventure-game
Hope that helps!