Mike, if memory serves me correctly, that Honda has fuel injection. That being the case, it sounds like your fuel pressure regulator isn't functioning properly. An electronic fuel pump is located in the gas tank. That pump gives off a constant pressure and that pressure is then regulated to the fuel delivery rail by allowing some of the pressure to bleed off back to the gas tank on a return line via the fuel pressure regulator. The fuel pressure regulator is vacuum operated, so take a look along the fuel delivery rail for a sort of cylindrical apparatus with a vacuum line attached to it. Insure that the vacuum line is not cracked and leaking. Repair or replace that first if it's bad. You can check vacuum leaks with a hand vacuum pump available at most auto parts stores. Just make sure that line isn't leaking. Vacuum leaks can also present themselves as rough running (not smooth idles). The fuel pressure regulator is an easy replacement if it's bad. The fuel rail on your car is a shiny aluminum stick that runs along the back side of the engine between the engine and firewall. The fuel pressure regulator is located on the drivers side of that rail. You also may want to have a dealer run fault codes on the car. That car is equipped with an OBD interface under the steering column. It looks like a old style computer cable terminal connector. Sometimes the fuel pump is faulty, but in order of least expensive repair, the vacuum line to the fuel pressure regulator is only a few dollar. The fuel pressure regulator is maybe $60, but the fuel pump is about $500 for someone to replace it. I am enclosing a vacuum line map that will show the fuel pressure regulator.
Let me know how it works out.