Your biggest problem is sharing a console with someone else. That's not a good way to be and that's also the main reason you're out of HDD space. Other than that, just play game Y while game X is installing, or do something completely different.
People usually have the whole 500GB or more to themselves and I only play one game at a time anyways. Maybe that's part of the problem, too. You have too many games going on simultaneously, and your attention is divided, so you just kind of tune out instead of following through with one story at a time and really getting into it. Instead of experiencing a journey from start to finish, you go in intermittently like "where was I again, what was this about again?".