This same thing bugged the crap out of me for the last couple of years. It took a while, but I think I've pretty much taken care of it in my own swing. I was probably worse than what you describe as I would even set up right of my target hoping to play that normal ball flight, but I would then start the ball at the target and have it draw more left. Anyway, I would have the club face a little closed and swing too much to the right causing the pull draw.
I tried to lay down the club at transition then tumble to get more on top, but this was way too inconsistent. I finally have found a pivot that works really well at straightening out my ballflight. It is utilizing the Kwon/Como pivot information with the key for me being when my lead arm is about parallel on the backswing as the center of pressure is moving to the left, I also open my pelvis to start the rotation; this also has to happen without lateral movement for me. This move causes my club to lay down on its own. From there I just try to rotate the torso with the feeling of hitting the ball with the right side of my chest (and I mean literally if the clubhead were imbedded in my right pec). I also need to have somewhat of a pitch elbow and feel the hands getting out in front and left along with the torso feel. This is about the only way I have been able to somewhat automate the tumble.
I don't know what it looks like because I haven't filmed it, but this is the best and most consistent I have ever hit the ball. My only problem with it is that I can't go after the ball over 80% swing speed without slipping back into the old swing.
I don't know if you need most of these elements in your swing, but I know I needed it as everything else I tried did not last. Hope it helps. Best of luck.
