I had an FE with that problem and I was told the issue was shutter magnets. This is from a trusted repair guy. He told me he could fix it for $175.00. If the camera was set on the automatic priority setting you may be having the same problem in the FM.
My Nikkormat will also stick open on the slower speeds just like you are describing. If I turn the knob to B the shutter will close. I dont know how healthy this is for the camera however.
Try fresh batt's. Try the shutter without batteries. Try it on it's highest shutter speed and work your way down to the slowest and see where it sticks. This info will help when/if you send it in for repiar.