FWIW, in the CFL (that's the Canadian Football League for any of you whose version of the world ends at the US border 🙂), these are the rules for overtime:
Each team is given an equal number of offensive possessions to break a tie. A coin toss determines which team will take possession first. The first team scrimmages the ball at the opponent's 35-yard line and conducts a series of downs until it scores or loses possession. If the team scores a touchdown it must attempt a two-point conversion (I'm not even exactly sure what this part of the rule accomplishes; it was changed this last year. My dad, who watches a lot more CFL than I do says that it's a good rule. Again, not sure why.)
The other team then scrimmages the ball at the opponent's 35-yard line and has the same opportunity to score. After the teams have completed their possessions, if one team is ahead, it is declared the winner; otherwise, the two teams each get another chance to score, again scrimmaging from the other 35-yard line. If there is still no winner after this second round, the game is declared a tie during the regular season. In a playoff game, the teams continue to attempt to score from alternating 35-yard lines until one team is leading after both have had an equal number of possessions.
(And while we're at it. You guys really need to look at our kick return rules. That fair catch takes away a really exciting part of the game. In our game, there is a five yard "safety zone" around the kick returner until he touches the ball. If a defender enters this zone before the ball is touched, there is a "no yards" penalty. Otherwise, the ball on a kick is live. Lots more fun. A LATE EDIT: For clarification, the kicker is exempt from this 5 yard safety zone and can recover his own kick. That part of the rule stops a kick receiver from standing over the ball to either run down the clock or giving his blockers time to create a formation. He has to make a play on the ball before the kicker gets there or risks losing the ball.)