Hey Ari,
By any chance did you buy the In-Cell LCD version of the screen? If so, that could explain your issues. The thing is, the original screen is an OLED, where each pixel on the screen generates its own light. The older LCD technology is like a valve; it can only open and close to let light through; it doesn't create it's own light. So there has to be a light source for an LCD display; that would be the backlight; basically a set of LEDs that light up a white sheet of plastic to evenly distribute the light over the entire screen.
The problem here is that backlights take a lot more power than an OLED display, so the LCD replacement screen is drawing more than the original OLED screen, which could certainly explain your heating, as well as the battery drain. What's your battery health at? Your phone's old enough it could be getting to be time for a replacement, as the original may no longer be holding a full charge, making the situation worse.
If you bought one of the OLED versions, whether hard or soft, then there's definitely something else going on; as @smashedit has suggested, it could be a connector not quite 100% in place, so rechecking your work would be worth your time.
As long as you have it apart, it wouldn't hurt to plug in the old screen and verify the problem doesn't happen with it. Keep in mind that iFixit has a lifetime guarantee on the screens, so if it's not working correctly you shouldn't have any trouble getting a replacement under warranty.
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