I have the same exact problem as you!
Mine started after I decided to crack open the case on mine to fix the rattling loose screw issue I've had for 2 years.
FWIW, I bought a used Kindle 3 Keyboard (wifi only) and tried swapping screens. No luck!
Here's what I've been through so far in detail:
https://www.mobileread.com/forums/showpost.php?p=2367694
So mine is 2+ years old and very much out of warranty (not to mention I cracked it open). I called support and got a nice guy who couldn't help me and suggested I put it back together and use their trade-in feature to get about $15 toward a new one.
Did you send the old one in?