I can think of several things off the top of my head from bad gas to clogged filters. Several things tho will pinpoint the area of concern. I make several assumptions from your question. 1 This only happens when cold. 2 Once its running it runs fine. 3 you checked and changed the obvious air filter, gas filter and refilled the tank at some time after the problem happened. If you didn't then also check those.
Next I would look at the temp sensor and see if its working. You probably have the V6 so its on the middle of the engine near the fire wall. Then I would remove and check the cold start injector. Its on the left side of the intake manifold. Has a bolted hard line from the injector manifold. Internally they get pretty clogged due to the dirt in this area.
Lastly check the emission system hoses for cracks or holes. Especially where they enter the engine. Take one end off follow to the other end and see if you can blow air thru them. Then check and ensure they are open going into the manifold. Again the environment they reside in can be dirty and clog the ports.
Could be many reasons. If the check engine light is illuminated, check for codes first.
A bad fuel pump may make the engine sputter. The engine may cut out and come back on, at first. Eventually, the engine will not run at all.
The best way to determine why the engine sputters and dies is to inspect the spark plug after the engine dies. There are two most likely reasons. In the first example, the engine will most likely just sputter and die. In the second, the engine will generally blow black smoke when it sputters. (1) The spark plug greyish is dry: It is likely that the fuel filter is too clogged to allow flow of enough fuel to keep the engine running. If the fuel filter is not clogged, then the carburetor needs adjustment. (2) The spark plug is coated with black carbon or is wet (or both): The engine is running rich and is flooding. Likely causes are a dirty/faulty spark plug; a dirty/clogged air filter; or the carburetor is set too rich.
not enough oil in the engine.
The 97 Cavalier could be sputtering because of loss of fuel pressure, or weak spark, among other things. Check these two first to start troubleshooting the problem.
Check the battery and the connections to the engine.
Bank 1 is the side of the engine with the number one cylinder, in your vehicle it is the passengers side. Sensor 1 is the first one coming from the engine, before the catalytic convertor, in the exhaust pipe.
Causes. Water in fuel/bad fuel. Check. Low fuel pressure. Fuel filter stopped up. Check these first and if you don't find the problem in this area. Then Please ask your question again.
Air in the water lines to the upstairs. Not enough water pressure.
One of the control devices is malfunctioning on the engine, and it needs to go to a shop for repairs. Simple things first though. If the light just came on, check to make your gas cap is on correctly, leaving it loose will cause the check engine light to come on. The light will reset itself after awhile of normal driving. If it does not have your local autopart store or shop read the code. That is the simplist way. Does the engine sputter while driving? That is an electrical problem with the spark plugs not firing when they should or not at all. Change the wres and plugs, check timing. That is what happened to mine
Fixing the problem which causes it would be the first step.
Try to find a fuel pressure gauge. The fuel pump sounds suspicious from your description.