I know you said no auction, but here's an auction concept for you anyway
An auction / snake hybrid, where the auction sets the draft order of the snake draft, rather than leaving it to pure chance.
First pick goes up for bid, where the bid is how much of your FAAB budget you're willing to sacrifice for the #1 spot. This format could include the 3rd round reversal, making the top pick less valuable (or even the end of round 1 more coveted), but my league has always done this based on a standard snake format.
Winning bid comes in, that person is locked in at the 1-spot,
less their FAAB bid for the season, and will proceed to make the first selection. Keep on going down the line until you reach the point in the 1st round where no bids come forward. Skip that spot and move on to the bidding for the following spots, in case there are spots further down that are more coveted, even without knowing who will be available. Once you're done trying to coax FAAB bids for all the selections, you will then do a random draw to assign the open draft slots to the remaining players that didn't want to spend any FAAB to determine their draft slot. Proceed with the snake draft per usual from there.
My league generally seems to
prefer the casual aspect of a snake draft over an auction, but this bidding concept to determine the draft order of the snake has been well received since its inception. No champion has ever elected to revert back to a luck of the draw snake. It at least gives teams some control to go after their top pick, while maintaining the ease of a snake draft.