How do I avoid chunks of butter in non-homogenized milk?
Yes, this is the way it is. The point of homogenization is to prevent the chunks of "butter" (it's actually kaymak, halfway between cream and butter). If you don't want them, you should drink homogenized milk. 

As for "too slowly", yes, you are drinking it very slowly. Normally, you should end a carton within 3-5 days of opening, regardless of the end date on the carton (it assumes that a carton remains sealed). If it's raw or standard pasteurized, you'll know when it spoils, so you can still continue drinking until it curdles. If it's high temp pasteurized, it won't change its taste and appearance when it spoils, it will only turn slightly bitter, but most people don't notice the difference, especially if not drinking it pure. 

You could, in principle, add emulsifiers and beat it smooth again. But you will change the taste and texture to a point where it would be to most people's taste to either drink the semiseparated milk or to buy homogenized.