Hi Michael,
It seems as though the digitizer in your phone is faulty (as you probably know) and needs to be replaced.
Here is one source for replacement parts for your phone. (Select the part suitable for your particular model number). It is not a recommendation to buy from the seller it is only to show what is available and what the cost is.
http://www.etradesupply.com/catalogsearch/result/?warehouse=0&q=galaxy+mega
Here is a link which shows how to replace the digitizer in your phone. It is in German but the video is self explanatory. It may be easier, (and less stressful and perhaps in the end cheaper), if you are going to do it yourself and do not have all the required tools necessary, to replace both the LCD screen and the digitizer. Then you do not have to separate them, just to replace the digitizer.
https://www.youtube.com/watch?v=edURB0NNdDk
Hopefully this is of some help.