If the screen was an cheap-repleacement part then it could be not suitable for your device. The good practice is to manage a hard reset of your phone after any replacement in order to get a new drivers adjusted to your component. Try that and then you should consider an PCB damage - not only the connector but whole ICs itself. The first thing that goes off when some one not provided goes into the device is an backlight IC, but in your case it could be just touch circuit.
Try a new display after an hard reset of device. Then go back to this forum and get better answer than mine.
Ps. After reading your post over again I would suggest to check for water damages. If the screen stops working after few hours it could be some dirt/corrosion on the motherboard near chips/capacitors