No, the will likely board early, so they can be introduced to the flight attendants and take their seats. They will not separate 2 minors traveling alone. In my opinion, it is worth saving the money. If you are still concerned, call Delta, explain your situation and I am absolutely sure they will accommodate your needs. The way parents are these days, airlines can't afford to mess around with things like this.
Also, Delta does a really good job with UMs.