You are absolutely correct my dear Watson. You are leaving your grip to open the club face. You also sound as if you might be new to the game.
I am relatively new as well but I have become consumed with strightening my shots as well as breathing, eating and sleeping golf.
Is sounds as if you have a "weak" grip (golf term) which means that your hands are more on the lower side of the shaft. You are correct in deducing that if you more your hands to the top of the shaft ("strong" grip) your ball will begin to more straighter and evern to the left.
I have some suggestions:
1: get lessons (at least 2-4) work on set up & grip They are worth it
2: read what Greg J. Willis says on this site (very informative and
revealing)
3: play from the left side of the teeing ground and
4: lastly play as much golf as you can.
Hope this helps

