Generally a sign of dead batteries in the FG, and some other cameras like other electrically controlled Nikons and Olympus, is that the shutter fires but the mirror stays up and the cycle does not complete, so you can't fire it again. Thus, if you see the mirror stuck up, it doesn't always mean sticky foam (dead batteries are much more common IME).
On cameras with a mechanical speed like the FG, you can complete the cycle by turning the shutter speed to M90 or B, and the mirror will come down and you can wind on.
As others said, setting the shutter to B doesn't necessarily preserve the battery, but pressing the wind lever against the body should turn the meter off and stop any battery drain. If your batteries die anyway, it's possible that the camera is draining the batteries in the off position (I don't know if this is likely for the FG) and you might need to remove the batteries between uses.