Some other ideas (not my original ideas, just passing along ones that sounded better than what we have now):
Idea #1: When the 4th quarter ends, if the score is tied, there's a short break, and then play resumes "as-is", using normal football rules. So if Team A has the ball and it's 3rd down, if the clock runs out in regulation then it's Team A's ball at the same spot on the field, and it's 4th down. It's sudden death, no game clock - just play until one of the teams scores.
Idea #2: Same rules as the NFL, except both teams get the ball at least once. If the team that gets the ball first scores, the other team has to beat that score on its first possession or they lose. (So if Team A gets the ball first, and scores a field goal, Team B needs a touchdown. If Team A gets a touchdown and kicks an extra point, Team B needs a touchdown and a 2-point conversion.) After that first set of possessions, if the game is tied then it's simply sudden death.
I also like the idea posted earlier in this thread, about the winner being the first team to score 6 points.