Rounded shoulders is a common phenomenon in people who work their front side of their bodies (chest, shoulders), but not the back side. Couple that with sitting most of the day, you got a recipe for achy shoulders.
To counteract this, you need to start focusing on your backside, specifically, your rear delts, mid and lower traps. Strengthening and tightening these muscles will naturally pull your shoulders back over time.
A general rule of thumb is if you are pressing in a horizontal plane (a bench press, incline press), you should do a horizontal pull, such as a barbell row or a cable row. If you are doing a vertical press (shoulder press, barbell military press), you should do a vertical pull, such as chinups and pullups. At the very least match the volume of push and pull. So if you did 25 total reps of pushing, you should match that volume with pulling to balance out your shoulders. However, since you are currently at a deficiency, I'd focus the next 2-3 months on doing more pulling than pushing. So if you did 25 reps of bench pressing, follow it up with 50 reps of horizontal pulling.
You should get most of your volume through the compound lifts, such as a barbell row, dumbbell row, t-bar row, cable row, pullups, chinups, pulldown, etc. However, it is also a good idea to isolate these smaller back muscles, as they often get favored by the larger muscles of the back, such as the upper traps and lats when doing compound back exercises.
The easiest way I've implemented this is to do Band Pull aparts(see athleanX video) or band facepulls between all my pressing movements (all bench press variations as well as overhead press variations). For example, do a set of bench press and during your rest, superset it with a set of 15-25 of band work. By the end of the workout, you'll get over 100 reps of quality back work. And on top of that, do your compound back work. I know it seems like a lot of volume, but your back, like your legs can handle a larger workload since so many muscles are involved.
But remember for the band work, you want quality reps. Check the ego and be sure you get as much out of each rep as possible. For example, if i'm doing a 5x10 overhead press, I'll superset it with band pullaparts 5x20. I can do 50+ reps on a single set of a 25-40lb band, but since I'm accumulating volume over more sets, I'll do sets of 20 instead. This ensures that I'm still taxing the muscles, but not to the point where it takes away from the primary compound movements.
Finally, I'd start taking your vertical pressing more seriously. Look up how strongman (Brian Alsruhe has a phenomenal video series on overhead pressing technique) overhead press as they are by far the best pressers of all strength athletes. If you do this with proper form, it actually works your upper back and thoracic region to a tremendous degree, when doing using proper technique(shoulders packed tight, core braced, glutes engaged, etc.).
Try this out and be patient. In 2-3 months you should see improvements as long as you keep progressing on your back work.