I had a similar problem with my 2003 Grand AM. Replaced fuel pump, fuel filters and more, nothing seemed to work. It took over a year to finally get it fixed. The final solution turned out to be the ignition switch. I tried moving the key around in the switch before and it did not help, but replacing it fixed the problem. It has been a year now with no further problems. I noticed one difference before and after the fix. At times, before the fix, the fuel pump buzz could not be heard when the ignition switch was turned on. Since the repair, the pump always comes on with the switch. You have to be pretty quiet to hear it because the electric pump is in the gas tank.
Source: Personal experience