The current NFL overtime rules are unfair, as both teams are not guaranteed a possession. If the team that gets the ball first scores a touchdown, and they are playing against a tired defense, the game ends without the other team getting the ball.
While there is no way to make this completely fair and the team that goes second has an advantage, I think the following is a more fair way to decide the winner. Here are two options:
1. Play a normal football game, with each team getting a possession. If the game is tied after both teams have one possession, continue playing with both teams getting a second possession. If the game is still tied after a second possession, have both teams get a third possession. Continue this until one team is ahead after both teams have the same number of possessions. This is similar to extra innings in baseball, where both teams get equal opportunity.
2. Give one team the ball at their own 20 with 4 downs on each series. The other team would then get the ball at their own 20 with 4 downs on each series. Both teams would go in the same direction. If the game is tied after one possession for each team, repeat with a second possession for each team. If the game is still tied after two possessions, repeat until one team is ahead after both have the same number of possessions. The teams would alternate going first to minimize that advantage.