I think we're looking at a fuel pressure issue here... either it is not getting enough fuel, or for some reason the computer thinks it's getting too much fuel and is leaning out the mixture as a result.
Your symptoms are typical of one of two things. Either low fuel pressure or low fuel flow, or too low of an air intake calculation.
I would start with the fuel theory. You'll need to install a fuel pressure gauge using an adapter on the engine side of the fuel filter. Key on, engine off pressure should be 38-44psi. Key on, engine running pressure should be 31-37psi, but should increase to 38-44psi if the vacuum hose at the fuel pressure regulator is removed while idling. If you can get it to act up while monitoring these pressures but don't see a decrease in pressure at that same time then this is probably not your problem. If you DO see a decrease in pressure then you'll want to monitor the voltage at the fuel pump while this happens, and if the voltage doesn't decrease then the fuel pump is at fault.
Another possibility would be the MAP (Manifold Absolute Pressure) sensor mounted on the firewall on the passenger side of the engine compartment. There is a vacuum hose going to this and the computer uses this sensor to determine how much air is entering the engine (lower vacuum means more air volume). If this sensor is plugged up or malfunctioning it can cause similar symptoms. Remove its vacuum hose and make sure there is plenty of vacuum at that location. Make sure the hose itself isn't plugged and make sure the sensor itself isn't plugged up either.
Another possibility is the Oxygen Sensor
. If it is 'biased rich' the computer could be leaning out the fuel/air mixture to compensate for what it thinks is a rich running engine. If it is not really running rich then this can cause the same driveability symptoms you describe.
The operation of both of these sensor can be verified with a compatible scanner, but this requires a scanner capable of communicating with the older Toyota systems. Not every shop will have this.