You could always subscribe to mlb.tv premium. They allow you to choose the broadcast you prefer. It also costs 119.00 per season for that option. If you are watching on MLB Network, they choose which broadcast to feature. If you are paying for the Comcast MLB package, they also choose which broadcast to feature. For choice, go to mlb.tv premium. I use it to watch Cardinals games and always see my Cardinals broadcast.
Another option. Buy a blu-ray player. Mine has web connection. With it I could watch the mlb.tv games on my TV. You would still need to subscribe to mlb.tv premium. It might come with the mlb app or you could download it onto the blu-ray player. You also would need to live outside the Chicago White Sox territory in any case. If you live inside their territory you should be able to see your games on TV. Since you can't, I suppose you live elswhere.