Dear Jeremy,
In my view, using the dominant hand to power the stroke is the problem that causes pushes to the right. Actually, some of these are probably cut-strokes in which the toe is lagging too far behind in the forward stroke -- this is also a miss to the right due to the dominant hand.
When you putt left-handed, you probably tend to be a little confused on which hand should provide power, so that the dominance struggle between the hands sort of washes out. This results in a stroke WITHOUT so much corrupting influence of a dominant hand, regardless of whether it is right or left. If you "learned to putt left-handed" the same way you now putt right-handed, the problem would likely just resurface in a new form.
Hand dominance is more of a "hit" stroke that relies upon hand-eye coordination to manage delivery of the putter head into the back of the ball. A number of "habits" encourage this, as does most putting instruction. The confusion seems to come from not separately understanding power and line. People who rely on the dominant hand for "touch and feel" are the ones who experience problems with the line, and they fear losing "touch and feel" by blaming the dominant hand for this problem.
My teaching is: You don't lose any "touch and feel" by getting away from hand dominance and handsy control in the stroke, and you definitely gain in precision of line by doing so.
How's that? Because "touch and feel" for distance control are illusory. What gives you excellent distance control is relaxed, stable tempo and instinctive backstroke (based on sense of green speed and targeting). Excellent distance control "happens" with good timing and an instinctive backstroke regardless of your "touch and feel." What USEFUL "touch and feel" really amounts to is a relaxed body-state that is set to allow the instinctive backstroke to kick off the well-timed forward stroke. But even if you are not relaxed, if you manage to get the backstroke right AND the forward stroke is timed the same as always, the result is excellent distance control. It's just physics, believe it or not. Apart from the brain's system for generating the instinctive backstroke that is called for by the putt and the green speed, "touch and feel" or distance control has absolutely nothing to do with the individual golfer other than the golfer abiding by the same tempo the brain used to generate the backstroke. (Don't mess it up by getting cute or trying something "artistic.".)
You won't believe this unless you try it for a while, and to try it you first have to learn how to do it. I would suggest starting with my tip The Core Putt. Then try the one's about Tempo and the Gravity-Flow Stroke. Once you have the tempo and the instinctive backstroke inculcated, then you are in a position to focus on aspects of the form of the stroke that have to do with missing the intended line of the putt.
What you should find at this point is that the dominant hand is the problem, causing unwanted stroke path and face angle changes that result in missing the intended line. The more you learn the timing of your stroke, the less reliant you are on a dominant hand for distance control. As your distance control becomes better, your line also becomes better, but the lingering culprit is the dominant hand. Ultimately, tempo is the basis not only for distance control, but also for line control.
So what to do? A. Stop fretting about losing "touch and feel" without your dominant hand in control of the stroke. B. Start focusing mostly on your tempo. C. The more you learn your own tempo, the easier it is to make a straight stroke without handsiness. D. Observe how distance control and line both boil down almost entirely to sticking to your tempo.
I'm sure you'll need some more pointers along this way, if you choose to pursue it, so don't be shy about calling on me -- either on this Forum, by email, or by phone. Glad to help if I can.
Cheers!
Geoff Mangum
Putting Theorist and Instructor
Geoff Mangum's PuttingZone
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