The issue of muscular balance, especially for shoulder health, is really getting some good attention now. Compare the weight you can pull, in the horizontal plane, to what you can push in flat bench.
I went through a similar process recently and began to perform seated rows with elbows high, using the same wide grip as with bench press. My posture has definitely improved over the last 6 months!
Now I can pull much more than my lower back can hold, with my upper torso unsupported, so I used a seated row machine with a chest pad.
Think about this: how much could you push in bench press without the bench to support your back? How can you develop this same strength and balance with cable rows or T-bar bent-over rows without chest support?
This really helps you concentrate on the pulling action. Using a machine with a chest pad is, essentially, like reverse bench press. The machine at my gym uses individually articulated arms, preserving a more natural, complex movement.
I also place these pulling exercises earlier in my workouts, before horizontal pushing, to favor the growth of upper back/rear delts and improve the balance.
Good luck - Nylo