After the 6th card is dealt (6 player, 3 have folded) the dealer mix in the dead cards and shuffles the deck (the dealer forgot that we had 1 more card coming).
What should happen now?
There is only really one thing that can happen. Reshuffle, burn and give everyone their 7th card.
The only other option, and this is only because it's a home game, would be to have the players who have folded tell the dealer discreetly their cards, remove them to the muck and then just shuffle whats left. Assuming they can all remember their cards. At least you won't be re-dealing other players cards. This is kinda silly in my opinion but as a home game you could do this.
If any cards in the deck have touched any cards in the muck, the hand is void. No reshuffling, no attempt at reconstruction, everyone gets back the chips they had at the start of the hand, the button doesn't move, and you start the hand over like it never happened.