Driver heads have gotten huge, but I still see plenty of heel and toe hits. The reason is, most golfers swing the driver too hard, with either the hands and arms or the hips racing out of sequence on the downswing. Either way, the club gets re-routed, leading to off-center contact.
If you yank the club down from the top with your hands and arms, you move it away from your body. This causes contact on the heel because the clubhead swings across the ball from outside the target line. To fix heel hits, you need to keep the club to the inside. Close your stance, and focus on hitting the inside portion of the ball, swinging out to the right (photo).
If you slide your hips aggressively toward the target to start the downswing, the club tends to drop too far to the inside. You've heard tour players refer to this as "getting stuck." It causes contact on the toe because you don't fully extend the club out to the ball by impact. To stop toe hits, you have to get your arms back in front of your body sooner. Try to make a smoother transition with your lower body.
Whatever your fault when you go hard with the driver, remember that making good contact is the biggest power issue. More speed means nothing if you don't hit it flush.
Is your 3-wood working hard enough for you? It's probably a good option when you need control over distance off the tee, but you also need a wood you can hit consistently from the fairway. You might be better off dropping your 3-wood for a 4-wood, which has another couple of degrees of loft and a slightly shorter shaft, making it easier to hit off tight fairway lies and almost as long off the tee.