Hours of searching for this fix and you finally have it!

Works like a charm. I should mention, my mom had the same issue with her phone (same model) and I had her do the same thing and it worked again. Very reproducible! A million thanks to you kind sir

!!!
The details: screen Timeout set to 10min, keyboard timeout was set to "same as screen timeout". Changed keyboard timeout to 1min and the flickering stopped. SoftKeys were fine, keyboard backlight also fine.
Then, I changed the keyboard timeout BACK to "same as screen timeout" and the problem
did not come back. So, whatever it was, just went away. I'm going to keep it as is. I know it's not an app conflict because my mom has no apps installed (besides the default ones it came with).
I'm curious to see if (with the settings I have now - the ones that caused the problem?) the problem comes back after a reboot. Anyway, hope this helps. Note that the problem was not solved with a reboot.
I had called Sprint technical support and explained the issue thoroughly - he was quite certain that it was a hardware problem (probably wiring) and I should take it in to be replaced. Weird. I should probably call back - save them some money