iPhone 5S screens work on 5SE phones but you need to make sure that the LCD is original. Or atleast that is what I have noticed. As for your phone. look at the connections very closely and unplug the battery to make sure it fully turns off before you turn on the phone with the original screen again. hope it works!
Yea to replace it you just need two screwdrivers and a plastic prying tool. Look up tutorials on YouTube and that should fix your issue. I do recommend that you do it soon though because swelling batteries tend to keep swelling and can crack your screen.
I just had this same exact problem and I too have been working on phones for some time now. When I finally realized that the LCD worked without the digitizer then I took a close look at the digitizer connection and sure enough it was busted. The top of the connector on the motherboard looked like it was pressed down a little too hard and the metal pins at the top looked partially out of place. So that could be the same issue you are having. I'm going to try and solder the connector back together and see how that goes. Ok so soldering didn't do anything but the next day I plugged it all back in and placed some pressure on the digitizer connection and it booted up fine. I bumped the phone a couple times to make sure that it wouldn't come loose and haven't had a problem since. The image attached is the connector and notice how the top of it is a little crooked.