My new old 96 Ford Contour is sputtering when accelerater is pressed

Change the fuel filter if it hasn’t been done recently. This is the easiest trouble shooting step (providing) the fuel filter is in-line and not in the fuel tank.

If changing the filter doesn’t help, have a fuel pressure test done to the system to possibly locate other faults. (like a weak fuel pump)

All that is provided the problem IS fuel supply related.

Other than that, there MAY be an electrical glitch somewhere.