I also have an Evo Shift 4G and experienced the exact same issue you're describing. The phone I used before switching to the Evo Shift was an HTC Touch Pro2 (a phone with a slide-out keyboard - pretty much exactly the same design as the Evo Shift) and it also stopped working because of this issue.
Basically, the problem is caused by a short in the LCD cable that connects the screen to the motherboard. Because of the phone's slide-out design, that cable has to run between the hinges that allow the keyboard to slide in and out. Over time, as you keep opening and closing the slide-out keyboard, that cable rubs against the phone over and over eventually causing it to short out. Another common issue also reported is that the hardware keyboard stops lighting up and working. This is also caused by the same reason (the data cable connecting the keyboard to the motherboard also has to run between the hinges of the phone)
Now if you're like me, you hate phones that don't have a hardware keyboard. I hate typing with software keyboards and I won't buy a phone unless it has a hardware keyboard. That being said, it is with disappointment that I share the following advice: If you want a phone that will (generally) be free of those kinds of issues, you need to stick with phones that are not designed with moving parts (flip phones, slide-out phones, rotating phones, etc.) since they all are manufactured with cables that have to run between the moving parts, which over time will eventually rub and short out.
As for me, I pay the $8 handset insurance and take the risk. I won't give up my hardware keyboard. I love them too much.
Return the phone, get a replacement, and buy the insurance. Good luck.