It's hard to tell, but is your driveway cut into the side of a slope or through a knoll? Is that the origin of the berms? Or, are they banks pushed up by a bulldozer when the driveway was built?
I don't think, from the pics, that you are working with a wide enough overall space for a driveway and ditch. Ideally your driveway surface would be the highest thing, then lightly sloped berms 3'-4' wide, then a ditch that is deep enough, ~ 2'-3', to draw ground water from the elevated drive and berms.
It would mean moving a lot of material to get to what I am picturing compared to the pics, but if you do several hundred feet each summer, eventually it gets done. Your backhoe and loader should be able to do that as long as you don't encounter big rocks or ledge. They can be dealt with too if need be.
An intermediate solution would be to cut through the berms every so often and direct the runoff through those cuts if there is enough fall away from the outside of the berms to work with.