Your ballcock has nothing to do with the flushing of the toilet (unless this is a one piece toilet). Your problem is most likely in the flapper, over time rubber can breakdown and cause your problem. When replacing the flapper make sure you replace it with the original flapper for your toilet and not a “fit all” with today’s toilets we find the original flappers work best. If you need more info go to our web site and look at the toilet breakdown pages to help you with the part #.
Good luck,
Scott
http://www.plumbingpartsdepot.com