No acceleration

Do you mean you push on the accelerator pedal and the engine does not speed up, or you shift into drive, press the accelerator, the engine speeds up, but the car does not move? What is the history that led up to this? You really have not provided any details or information that can be used to start a diagnosis.

That tells me even less, but "rough running" is a symptom we can work with. The new ignition switch parts are involved with the anti-theft system, and that is obviously working okay because the engine runs. "Plenty of fuel pressure" is not accurate. Many GM engines will not start or run if the fuel pressure is just a few pounds too low, but it also cannot be too high. That will cause an excessively-rich condition with black exhaust smoke, and possibly a rough-running engine.

Fuel pressure specs for a 3.1L in a Cavalier is 41 - 47 psi.
Fuel pressure specs for a 5.7L in a Caprice is also 41 - 47 psi if you have an injector at each cylinder, or 9 - 13 psi if you have two injectors in the throttle body.

If the rough running only occurs at start-up, and always clears up very quickly, a good suspect is a leaking injector that is flooding a cylinder when the engine is off. Typically that is accompanied by an unusually-long crank time because fuel pressure has to build up enough for the engine to start, and that occurs more gradually during cranking because system voltage is being dragged down by the starter. That makes the fuel pump run slower than normal.

Diagnostic fault codes were relatively primitive on 1995 and older models. There likely will not be any codes that will help with the rough running, but regardless, you do not need a code reader or scanner. With Chrysler's, you just cycle the ignition switch, then count the flashes of the check engine light. Obtaining fault codes on Fords is real involved and messy. On GM's, it is almost as easy as on Chrysler's. Look for the diagnostic connector under the steering column. It is a rectangular plug with two rows of terminals. There will be two terminals right next to each other in only one corner of the plug. Jump those two together with a paper clip or piece of wire. Turn on the ignition switch, then count the flashes of the check eEngine light.

Here is a better description of the same thing, and the list of fault code definitions:

https://www.2carpros.com/articles/buick-cadillac-chevy-gmc-oldsmobile-pontiac-gm-1983-1995-obd1-code-definitions-and-retrieval-method