this is what works for me....
you want to do the opposite of what you suggest - you want the bar over your centre of gravity at all times to avoid excessive layback. If you have the bar forward, you will want to incline bench press it, or otherwise you end up putting a lot of strain on your shoulders.
Take the grip just outside shoulder width. Get the bar on your front delts with your shoulder and chest up high. This will get the bar right up against your throat. Should be the same rack position as used for a front squat or jerk.
You need to get it back into this spot with every rep. The more forward the bar creeps, the further back you'll want to lean.
When you initiate the press, get your chin up high to give yourself room to drive the bar straight away without leaning back. Then when bar is past your forehead, drive your head forward and get under the bar, driving your traps up to lockout.
This will feel completely different - you will feel it more in your upper back and less in your shoulders.
same layback will happen if the weight is heavy, but if it's too much this puts a lot of strain on your facet joints, which if you hurt them is really painful.