When holding the bow at arms length the bow tips back toward me. Last night I put a heavier stablizer on it and immediately noticed a difference in my ablility to hold the bow balanced and it seemed that my groups were tighter.
You sort of answered your own question........
The main purpose of a stabilizer is to promote good balance before during and after the shot which will enhance your aiming , follow through and accuracy.
There is not just front and back in this equation either.......do not neglect the other planes of movement either. Side, offset and backweight are sometimes necessary to achieve proper balance on many of todays newer bows that are easily overcome with a long or heavy front stabilizer. (Small offset stabilizers aren't just for the target crowd)
The bow should be set up that it's not pulling in any one direction excessively while aiming and the limb tips should not travel excessively in any direction when fired. You need to find a good static and dynamic balance to get the most of your bow.
Watch (or have a buddy watch)your bows top cam/wheel to see what direction it goes at the shot........whatever direction it dives, counter that with weight in the opposite direction until it sits almost still when shot.
The bow should not pull hard in ANY direction at rest or when fired.
This could mean any combination of stabilizer lengths , directions and designs........every bow is different, and accessories will also play a role in stabilizer choice. Brands and makes can be recommended, but lengths, weights and orientation cannot without first hand experience with a particular bow set-up.