Welcome to Android Forums Jas19kt. If shaking the phone sometimes causes the screen to become responsive leads one to think it's a physical problem. It could be at the connector or anywhere along the power to the screen. It doesn't sound like a firmware problem but I'm only guessing.
If it's a phone new enough to be cost effective to repair or as you suggest you might lose some important documents, I'd suggest seeking a reputable phone repair shop vs attempting the project yourself. Considering the minute circuitry of a cell phone, it's not something I'd care to attempt. But that's me. Good luck in whatever you decide to do.