I'm not absolutely clear about what you are trying to do, but since you seem to know what you are after, here's a suggestion that might help a little:
Go to the top of your swing and remove your left hand from the club and put it near your left side. Then swing down using only your right side. Observe whether the club stays on plane as you desire.
Repeat the process but this time when you are at the top of your backswing remove your right hand from the club and swing down using only your left side. Again observe whether the club stays on plane as you desire.
The results might indicate whether you have a right sided problem, or a left sided problem, or both.