I am not an expert by any means, however I have had a very similar issue as you described and still deal with today. I work at a desk most days like yourself, though have moved to a standup desk which has made a big difference.
Personally I feel your diagnosis is reasonable. Basically its upper-cross syndrome or as Kelly calls it in his book, Douche-bag shoulders. You seem to be on the right track strengthening the rotator cuffs and stretching out the chest.
You could try the Best Shoulder Mob Ever and see if it works for you. Myself, I did not find the shoulder capsule mob helped, but I am also feel I my UCS had regressed quite bad before I realized what it was and started treating it. I found lacrosse ball massage work on the infraspinaitus and suprasprinatus, along with rotator cuff strengthening (wall arm raises, prone cobras, etc) really helped. If you Google Upper Cross Syndrome there are a lot of good videos outlining some of the exercises. Crossfit’s Journal also released an article on rotator strengthening exercises which was helpful too.
Its a long journey to fix this issue, but persistence does pay off. Do you have Becoming a Supple Leopard? If you don’t, buy it. I found that following exercise descriptions as outlined in the book allowed me to return to pressing, however I need to stress that A) I had spent about a month ungluing my shoulders with a variety of MWOD mobility and physiotherapy and B) I do a lot mobility work prior to my workouts to make sure I have shoulder mobility before tackling it.
Hopefully that helps and good luck