[Al Skierkiewicz]

Ether,
With a car of your age, it is possible that the filter bag around the fuel pump is clogged or that the fuel pump is just weak. Most of the injected vehicles pump a rather high pressure to assist with the injection. A mechanic will likely do a fuel pressure delivery test to see if the pressure is reaching spec. According to one site your pressure should be in the 35 psi range. Many vehicles regulate the pressure by sensing pressure and turning the pump on or off. Some simply bypass the fuel back to the tank. It is common to only hear the pump during the first few seconds until everything is pressurized. At that point the pump is still running, it is just not making a lot of noise as it is running against pressure. The inertia switch seems to be a one time only device which would work or not. The fuel pressure is the first place to look but be very careful. Pressurized fuel rails is the most often cause of engine compartment fire. Follow the instructions carefully. You do not mention that your dash lights are indicating a problem so if that is the case, then a code might not be displayed on a monitor device. There is always the possibility that the wiring has gone bad with all of the salt and corrosion under the car in northern climes.
My pump makes more noise in the winter than summer.