I just finished the game today and had the same issue, as well as issues with loading and such. My Xbox 360 is rather old, and slow loading times are not unusual, but this was more like it just stopped loading after a certain point (tables not appearing in the main hall of Skyhold, NPCs and other environment things not appearing, etc). Traveling to a different location and then back to Skyhold again reverted loading to normal, but none of my companions would talk to me. I had a few brief dialogue options with Leliana, Josephine and Sera, but the rest just said one line about the new Divine and then proceeded to greet me as if we were strangers, and no dialogue options presented themselves (including Cassandra, whom I romanced). After the brief conversation with the characters I did have options with, they were similarly locked out. I do hope this is a bug that will be fixed soon, and not something intentional, because it's rather jarring to have all my companions be so silent and formal when I went to great pains to befriend them throughout the game.
