Maybe the screen is encrusted in which case being so shallow you can tremie some muriatic acid down to the screen area and agitate it or brush it. Or just dump it in and rod away to get it to the water zone. Pump it a lot and then chlorinate it too. The worst is that you will eat up whats left of the screen in which case you were going to pull it anyway. If your neighbors get water then probably the water table is not suspect.
Groundwater and wells is a old standby book and gives formulas for acid addition and safety instructions for this unpleasant operation which is not often done on residential wells..