“I also believe each team should get a possession. They already set the clock for 15 minutes. If the first team with the ball scores then they should kick off and the other team goes. After each team has possession then the sudden death comes in.”.
That is the current rule for NFL playoff games. Currently both teams must have possessed the ball in playoff overtime for the sudden death to kick in. The exception to this is if the first team to possess the ball scores a touchdown. Then the game is over. The purpose of this rule is to prevent crucial playoff games to be decided by a team winning the coin toss, running a couple of plays, and then kicking a field goal. The first team has to at least take all the way to the endzone for sudden death apply before both teams have possessed the ball.