Hi @jofixit22,
That's a new one on me too, but it seems pretty definitive that the digitizer is flaking out on you. Of course, on an iPhone the digitizer is fused to the LCD, so your only choice is to replace both. Fortunately, the screens are easy to get and since they're an older model the price has dropped considerably; I saw them as low as $20 USD on Amazon; here's a representative one for $25.
AmazonSmile: Ayake for iPhone 5S/SE 2016 Screen Replacement with Home Button Black, Full Assembly Retina LCD Touch Digitizer with Camera+Earpiece Speaker+Sensors+Tools for A1533, A1530, A1528, A1518, A1457, A1453 : Cell Phones & Accessories
The 5S and 1st gen SE take the same screen, so that widens your choices even further. If you want to do your own search, you can probably find them even cheaper on places like eBay and AliExpress, depending on how long you're willing to wait for shipment.
The one I listed comes with the full compliment of parts so you don't have to do a lot of parts transferring to the new screen, other than the home button so your Touch ID will still work. Here's the guide you need to do the job.
iPhone SE Screen Replacement - iFixit Repair Guide
If you choose to get a bare screen without all the components it will save on cost at the expense of more effort to move them over from your old screen. In that case you'll want to use the guide that includes those instructions here.
iPhone SE LCD and Digitizer Replacement - iFixit Repair Guide
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As an example, here's a random photo of my workbench, where I used the Markup function to fill in the middle bit in red. Works great -- except for that quarter inch on each side. [image|2727249]
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