I think part of the confusion comes from using the term "believe." I'm not sure it conveys to the modern mind the fulness of what was intended. I haven't done much more than a quick scan of my Strong's lexicons, but their are the overtones of faithfulness, steadfastness, trust, reliance, etc.
The drunkards, homosexuals, etc. that Jesus references are those who habitually, unrepentently embrace their sin. A believer does sin, sometimes even those very things mentioned, but he resists the temptation, struggles with it, repents of it, and clings to Jesus as his Saviour and Helper and Redeemer and Advocate. It is not merely an assent to certain facts of history and metaphysics. It's rather like a betrothal, an engagement, looking to that great wedding day. How does one behave when he or she has declared their intent to marry? Do they not focus their attention and behavior on that relationship? Do they not cease to date others? And don't they engage in planning and preparing for their wedded life? And even in marriage, if a man flirts with the waitress, he may receive a reprimand or a night in the doghouse, but probably not a divorce. He apologizes and, if he has any brains, will behave more wisely the next time. If he has an affair, his marriage will certainly be in grave jeopardy. And, if he spends every night carousing with hookers despite his wife's pleas to stop, his vows and his ring won't suffice to preserve the marriage.
Let me use another illustration, if I may. I was following my wife's car to an event. At one point, I got stuck behind a slower vehicle and got caught by a red light. My desire, however, was to keep my wife in sight so I wouldn't get lost and so I could go where she was going. She, in turn, seeing my plight, turned off the road and waited for me to get free and catch up, then led me on to our destination. I didn't intend to get held up, didn't want to be at that light, but was stuck. In hindsight, I probably could have avoided the trap if I had reacted faster. The key here, is that I was trying to follow her and go where she was going. When the light turned green, I headed straight for her.
These are not perfect analogies, but I hope they convey some of what I understand the term "believe" to mean in this context.