We had the same problem. 1995 25 inch Sony bought second hand a year ago from a friend. The blinking problem started some time back, first on warmup, then got progessively worse to the point it was out and blinking on every few seconds. Thanks to everyone who posted, ours is working now.
On our tv, the adjustment is on the circuitboard on the back of the picture tube. The knob faces toward the front of the tv. so if you look at the circuitboard from the front side, the knob is at the lower right. The sharpness knob is on the chassis and easily accessed.
I turned the knob on the circuitboad a bit clockwise, nothing. We thought we were just out of luck, but when I turned it ALL THE WAY COUNTERCLOCKWISE my girlfriend suddenly shouted "It's working!"
Assuming that the pot is typical clockwise up, the problem may lie there. Our set has a big hole in the side where it was bumped when moving and was really dusty inside, so I suspect the pot may the true culprit. And dust. So try moving the pot with someone watching, turn it back and forth and see what happens. We are good for now.
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