Fixing overly strong driver grip

I've only been playing for about 3 summers and starting to get pretty consistent on the course. Ball contact and hitting it straight have fortunately come pretty naturaly without slice or hook issues.

As I'm starting to swing harder and really put my body into drives, I'm having trouble not naturally leaning towards a REALLY strong driver grip to keep the head square. I find my right arm wants to naturally twist outward at address and my right thumb is almost pointing back.

Irons and wedges are pretty good on the overlap, which I know naturally helps a correct draw swing and hitting the left rear side of the ball, but I just can't get comfortable wrapping my right thumb over the driver grip. Or woods for that matter. Feels like the club head is doing whatever it wants in my back and down swing.

Anyone have this problem or know drills to help? I'm generally driving the ball pretty straight but want to get to a more natural, higher draw flight path. Ball flight is usually fairly low and straight. Not stinger or worm burner, just not as high as I know it should be arcing so it lands softly. Big problem ironing out distance on my woods because they want to roll so damn far.

Not sure if I described that well but hopefully some tips out there as I know what I'm doing isn't "right". Best to keep doing what I'm doing or really try and change something?