We are in a 10 team league so we use the State Lottery Pick 4. (Any lottery will do)
Each team is assigned a number 1-0(10). Everybody has this info ahead of time.
Then, starting on a determined date, we use the 4 digit number drawn by the State lottery. If the number comes up 7336, then the team assigned number 7 gets the 10th pick, number 3 the 9th pick, number 6 the 8th pick.
This continues until all picks are determined.
Our state has two drawings a day, so usually within two days the order is established. (Make sure everyone knows you're starting with the afternoon drawing, if that's the case)
There could never be a question of it being fixed, and I personally like the suspense of how it unfolds.