What would cause a car to shake and sputter and lose power only under heavy acceleration but not while at idle or low acceleration?

Check your fuel filter before spending money on other troubleshooting procedures.

Had the same problem, it would idle fine but about 4000 rpm or heavy exceleration it would start to jerk. so i replaced my cam sensor.

Fuel constriction, ignition parts corroded, wet spark plugs. I had this problem caused by a corroded connection on the high-voltage side of the ignition coil. Carburetors could have a failing acceleration pump.