For what ever the reason it appears that your pump isn't supplying enough water. It could be a high lift problem. It also sounds as though your tank may be nearly water logged. I'd check the air in the tank. If you do have to replace the pump I'd consider installing a 1/2 hp submersible pump. If you have to replace the tank consider installing a Pside-Kick which includes a small tank which is all thats required with the Pside-Kick and it's included Cycle Stop Valve. This will also give you constant pressure, like city water pressure!
I doubt that the problem is the screens are plugged in the faucets and shower heads because you say that it also takes a long time to fill the toilet.