2001 Toyota Corolla Step on the accelorator and the engine

With gas prices going upto record levels, my wife and I had elected to use our older Corolla as a commuter car. As of a few months ago, we had noticed that the car could not accelorate when we stepped on the gas and sometimes it would stall out. To date we have changed the furel pump, spark plugs, had the transmission tested, used fuel additives, changed the front oxygen sensor, had a mechanic check it over and we are still up in arms. About a week ago, I had detatched the Air intake Hose from the "Air Filter Box" to see if, in some way, incomming air is being restricted. To my dismay the car engine stopped running and would not even start! Through process of elimination I had elected to try and restrict the air intake, by placing a dry towel below the air filter. Under this condition, the car functions at about 90%, but I can't help but think that there has to be a better way. What do you think is wrong with our car?