My thoughts that are maybe the driver you are using doesn't have enough loft. When you use a driver that has too little loft, you subconsciously open the face to add loft at impact, which results in a high and right shot. Most people then change to a lower lofted driver because they are hitting it so high which only compounds the problem.
As you are hitting your 3 wood with a slight draw you should be able to do the same thing with your driver but i would suggest switching to an 11 or 12 degree driver, which will allow you to make a more powerful swing, while still getting the ball in the air.