"My question is how is the best way to add the second ball of yarn and then immediately bind off the stitches?"
Knit to the marker and leave the first ball of yarn attached . Add on the new ball, bind off to the second marker ( this leaves the bound off stitches in the middle of the neck which you will likely pick up later for some sort of neck band). Now you can work on both sides at once, using a separate ball of yarn for each side.
If you had bound off with the old ball of yarn , that yarn would now be on the left side of the work and you would have to add a new ball to work the right side. This is just a different way of doing it.