To steal someone else's metaphor, try thinking of Twitter as a river. If you try to bottle it up or consume it all you'll drown. If, on the other hand, you dip in for a swim as you please, you might find it all bit more manageable. As the rush of tweets flows by you, just check out some, then turn your phone off and carry on. When you load it again, don't feel you have to go back and "catch up" or give it the constant attention you mentioned.

YMMV, but I also think the emphasis on "conversations" is a bit misleading. Lots of people do have conversations on there, but I personally love it for breaking news, link-sharing, jokes, and photos. If your frame of reference is a forum with posts, or an article's comments, or a real-life conversation... you're going to have a bad time.