Engine raises and lowers between 1500 and 2000 rpms.
I presume you mean it does this idling the engine in the driveway. When this happens is the coolant up to normal operating temperature, engine warmed-up in other words, or does this happen only when first starting the engine, coolant cold?
If you are talking about the engine surging, then try cleaning the throttle body…
If you mean you see the engine lifting up and down, then you have bad motor mounts.
Eithe replace your idle air control, or. If you dont have one, have your throttle body cleaned.
I had same problem on my similar-vintage Corolla one time. Discovered idle rpm control for coolant temp gadget failed. Works similar to a coolant system thermostat, wax expands/contracts with coolant temp, and opens/closes air-valve. On Corolla that part isn’t replaceable, have to replace entire throttle body. I mitigated the problem by blocking off that air-pathway, so now in unusually cold ambient temperatures have to press on the gas pedal a little to keep engine from stalling for a couple minutes until engine warmed up. Fortunately ambient temps that cold are rare in San Jose.
How’d you block it off?
Suggest not to try this yourself. Ask your mechanic if such a thing is possible given your car’s configuration.