The engine shut off after overheating, now it runs rough and the Check Engine light is on

The car has an overheating problem. There was a blown fuse but that has been replaced. It was for the fan that cools the engine, but the car still overheats and then dumps all the coolant. The other night the car overheated and then shut itself off after cooling down I added more coolant and the car eventually started again but the check engine light came on and now the car rides rougher than before and when idle at a light it shakes like it wants to shut off and the temperature gauge fluctuates, but I don't think it's accurate any longer. The check engine light sometimes will now flash and once I can get the car up to a speed it feels comfortable at the engine light will then go back to on, but non flashing warning. The cooling fan sometimes kicks in but will make an awful sound. It's possible my water pump is bad, is there a relay or a fuse for the water pump? And if so, where is it located? I know the water pump and outlet and gaskets need to be replaced as does the engine fan and the entire thermostat and seal and gasket need to be replaced as well and eventually probably the radiator too, but my husband wants to start with the water pump and hoses first. Can you offer any advice please?