First check the obvious - what shutter speed were you using?
If this were the FE or any of its descendents, the symptom would suggest a dead battery. But the FM series is at the far end of the spectrum from rocket science,. Combine that with the fact that the FM itself was the first of the series (about 1977 or thereabouts) suggests that you could be dealing with a mechanical wear problem rather than something dumb like a battery.
But the good news is that because the FM is your basic mechanical machine, it's probably repairable. I had the entire shutter assembly replaced in my FM-2 about 15 years ago. Suggest that you make contact with the Nikon repair folks to see what they can do. Even if there is a simple fix to the problem, given the age this model probably deserves a good CLA.