Good call with the backhand - try to hit some left-handed forehands to better engage that arm for your two-hander. While you're at it, also experiment on the practice courts with your leg drive. Feel your weight driving off your left foot as you swing to contact - driving more deliberately with your left leg could prevent the stroke from being dominated too much by the right arm. It's okay for your weight to drive onto your right leg as you swing, but use the left for the big push.
Another thought that might activate your left arm through contact could be to start the stroke with your elbows at least slightly bent and focus on extending that left arm out (straight) through contact so that you get a good swing radius through the hitting zone.
As for the wristy malady, maybe try hitting some forehands (on the practice grinder) while gripping with no more than your thumb, middle finger, and ring finger. This might force your wrist to be more passive though contact, but it will probably challenge you to get the stroke going just a bit earlier, too.
You might also get less wristy action if you focus on a full follow-through with your elbow. Turn your shoulders through contact so that the elbow of your racquet arm is pointed in the general direction of your target at finish (if it isn't doing that now).