I just DLed the newest FCEUX to test from their homepage, not sure if I can link it here. Didn't have any problems at all. Make sure there's a (FE2 rom name).sav in the sav folder(this is the in-game save), and the savestate should be (FE2 rom name).fc0 in the fcs folder.
I tried both saving and savestating, then closed the emulator, and both loaded up fine. Might want to check the options for directories and set it to load saves/savestates from those respective folders, since it might be trying to load from elsewhere.