I am no great expert on these cameras, I've poked at Nikon FE, FM, etc but never disassembled one fully to understand the mechanism. I have a couple of questions, to clarify the problem, (not to provide a definitive answer).
1. The OP says the shutter fires as if on B. On B, the shutter stays open and mirror up until the release button is let up. Do you mean that the shutter and mirror close when the button is let up, or that they hang up even after the button is let up, until the DOF lever is pressed? If so, does this hang-up happen even on the B setting itself?
2. Does the shutter actually stay open when it is hung, or does the shutter close but the mirror stays up?
3. It is helpful to understand the sequence of events in these mechanical cameras. Usually, and I am pretty sure this is true of the Nikon FM/FM2, the shutter button causes the mirror to rise and the aperture lever to stop down. When the mirror rises, it triggers the shutter to fire. The first "curtain" (actually vertically traveling blades for an FM/FE/FM2 etc) opens, the timing mechanism runs, and the second "curtain" closes and triggers the mirror and aperture lever to return.
It is informative that pushing the DOF button causes the cycle to complete. This suggests that there is some friction or hangup that is preventing it, which is overcome by force with the DOF button. However, that would make more sense if the shutter closes, but the mirror doesn't come back down. It's not clear that the DOF mechanism should stop the second curtain from closing.
One place to look for excess friction in the mirror/DOF mechanism is if the aperture stop-down lever is bent from dismounting a lens the wrong way.
It is possible to remove the bottom cover easily, and you can watch the cocking and firing sequence with the cover off, which is unlikely to reveal a problem if it's in the mirror box/DOF but might be useful if you can see what moves when the cycle is completed.