Minamyna, I really don't think you are overhandling your cheese and I'm not sure why it wouldn't stick back together, unless it isn't hot enough to stick back together. When I stretch mine, I take the chunks out and smoosh them around a bit, the centers are usually still kind of grainy as they still have unheated curd in them, so they go back in the hot whey. I smoosh a few more chunks, then go back to the first one. Eventually, the chunks are heated throughout and I work the chunks all together. I do enough that I have a ball of hot curd about three inches through the middle.
At this point, I am able to start stretching the cheese. If it starts feeling tough to stretch, it goes back in they hot whey (I keep the whey on the stovetop at a consistent heat so that it doesn't cool off). I stretch and fold the curd around on itself until it starts looking smooth and shiny, then I form it into a ball. There's always a "nub" on the bottom, so the ball goes back in the hot whey for about 30 seconds, then I take it out and put it on a flat surface for about another 30 seconds, then into cold water to help it firm up.
I'm obviously handling my mozzarella a lot more than you do, but it turns out good. Maybe you are not stretching and kneading it enough?