Remote possibility, is the self timer switch activated?
When you look directly down onto the top of the camera, there are two levers (switches).
One switch/lever, the on off switch, is under the winder and when you slide it to the right, you will see a red spot. If you can see the red spot, the camera is switched on. If you cover the red spot with the lever/switch, the camera is switched off.
The second switch lever, is the self timer switch. This is underneath the shutter speed dial. If you slide this to the left, you will see a red spot. If you can see the red spot, the self timer is switched on.
In short, if you can see two red spots at the same time, the camera is switched on, and so is the self timer. When you depress the shutter with both red dots showing, there is a time lag, and during this time lag the red light on the front of the camera will flash, then the shutter trips.
When the battery is low, some funny things happen with both of my F3HP bodies, one of them is sort of emulating the self timer, but not actually doing the self timer, which is electronic and can be seen by the red flashing light on the right side of the camera directly under the "F3" sign on the front top of the camera face.
If you know someone with an F3 body and the MD4 motor drive, see if you can borrow the motor drive for a few minutes. With the MD4 motor drive attached, the electrics for the whole camera are powered by the motor drive.
Are you able to trip the shutter using the mechanical shutter release?
If you have two red dots showing on the top of the camera, wind the film on, don't touch the shutter button, but instead trip the shutter with the mechanical release. The mechanical release is the funny looking lever on the right side, it is all black. You use a finger nail in the little groove to pull the mechanical shutter release away from the body. Then, when ready, you continue depressing the little lever, which will mechanically release the shutter.
If that works, then immediately wind the film lever to cock the shutter, then release the shutter again with the mechanical release. If you can do this repeatedly then I think the shutter and wind mechanism are working properly and the problem may be electrical, and probably the battery is the issue.
Andreas Thaler's above advice is sound, and agrees with my decades of use with F3 bodies.