Forums are an open area to share information so you can call your dealer, or pose the question here, and hope that some other photographers that have used this or similar equipment might pass on some thoughts that might help. I am one of the later.
Can you provide more information on:
1. Lenses you are using,
2. Shutter speeds,
3. ISO
4. Shutter release
5. Mirror up or not
The reason I ask, is the AFDIII, is well known for the mirror slap. I used the AFDIII for almost 3 years with a P45+ and had some similar problems when the camera was on vertical. I can't remember if your leaf back has the internal rotation sensor or not. But for some reason the AFDIII seems more prone to give some shutter/mirror vibration when the camera is vertical. Make sure you have mirror up and with that set, make sure you wait a sec or two to allow the vibrations to subside. The effect you see most times is part of the image may look like double vision or you may see double edges. In the years I used the AFDIII and then the Df, I rarely shot the camera without a tripod, it just became part of the deal. I also usually used the remote release, (coiled cord)
Lenses, the more telephoto you go, the more problematic technique becomes. Really with Medium format anything at 80mm and out, you really need to make sure the tripod/camera rig is all solid. And you might again try mirror up. The older Mamiya glass can be excellent but you really need to watch your shutter speeds, with medium telephoto on out.
Shutter speeds, I would not try to hand hold the AFDIII, slower than 1/125 or 1/60 especially with a 80mm or higher. Wide angle you might be OK hand held.
Consider getting the Mamiya remote release. This allows you to trigger the shot without hitting the camera's shutter button. These are available on B&H and have a coiled cord. Well made and can give you a lot more shooting control with slower shutter speeds. The IQ backs, have the ability to trigger the camera (focal shutter) from the LCD of the back, which I have found to be invaluable.
AF or manual focus? I moved from 25 + years of using various DSLR and 6 x7 cameras to the 645 AFDIII in early 2008. I never was comfortable with trying to manually focus from the viewfinder unless I was working in close. You can purchase a viewfinder magnifier that attaches to the eye piece of your AFDIII which you might want to try. These are often on eBay. There is no AF micro adjustment on the AFDIII, but the AFDIII, is hit three times type focus. YOU RARELY GET THE CORRECT FOCUS, on the first hit. This is also a well documented issue and was not fix until the DF+ and even then with a later firmware update. But with the AFDIII, you need to develop a tap, tap tap on the shutter button and you can hear the AF working each time, usually the last tap will be the good focus. The AF can be very good, but the trick is to realize the first hit is most often NEAR, but not the best.
I mentioned mirror up, but that really is critical on the AFDIII or most Mamiya bodies, even the latest the XF. Unless you are shooting fast shutter speeds consider using mirror up on most of your shots. Also, work with your lenses, as certain lenses seem to have shutter speeds that they just seem to get out of sync with the mirror/shutter movement. My old 105-210, was one of these, and no matter what I did, the shutter speeds of 1/30 to 1/45 tended to have blur, even on tripod.
Just a few thoughts to try.
Paul C