I finally have a working Atari Lynx!
I've had incredibly bad luck with them recently. I have one that I've owned for around 12 years. After having it packed away for a few years, I went to play it only to discover lines in the screen. I figured the screen somehow went bad, so I bought another on Ebay. It worked for a few months, then the screen died completely (or so I thought), although the sound worked. I decided to do some tinkering and switched the motherboards in the two systems. Sure enough, the screen that used to have lines was now dead (although the sound worked), and the formerly dead screen had the lines. So, I now realized that it's a motherboard issue, and both screen are likely still good. The culprit must be bad capacitors on the motherboards.
I decided to try once more, and I bought another Lynx for $40. The seller was kind of a deadbeat - he didn't ship it for two weeks. I had to open a case to get him to respond. He also assured me there were no dead pixels in the screen, and there are a lot of them. Since I knew I had two good screens, I decided to mix and match to see if I could find a working combination.
In the end, the newest Lynx motherboard coupled with one of my older screens has produced a working system with a good picture and sound (for now anyway). It only took three defective systems to create one working one.