What happens if the team receiving the kickoff in overtime drives the ball for almost 10 minutes (unlikely but possible) and then kicks a field goal as the 10 minutes run out? Do they win?
According to this article, the Field Goal would be winning the game. It covers pretty much what you have asked.
The rules also cover this situation:
There shall be a maximum of one 10-minute period, even if the second team has not had an opportunity to possess the ball or if its initial possession has not ended. If the score is tied at the end of the period, the game shall result in a tie.
So if the time runs out the game is over.