Hey, gotcha. That's actually very common with the iPad mini digitizer replacement. So that would actually have nothing to do w/ the lcd. If its ever a problem w/ touch on your device it will be related to digitizer and any problems related to picture is lcd.
Most likely one of two things is going on. Either your replacement screen is faulty or your replacement screen didn't come with the protective cloth tape strip at the bottom. If you still have the original screen and look at the bottom inside of it, near the home button hole, you should see a thin black cloth like strip that goes along the width of the glass itself. This helps against what's called "ghosting" which is what you're experiencing. Its kind of weird to explain so hopefully that makes sense. If not or if you have any other questions don't hesitate to ask