Rock the grates with short choppy strokes back and forth, just enough to shake the ash through the grates. This won't allow the coal to drop and wedge between the grates.
It takes a little practice, but I'm sure you'll be much happier.

We try to be very careful about how much we move the handle back and forth. How many degrees off center would you recommend? We think the problem may be in the wide variation in the size of our lumps of coal. Is this common?

That's a common occurance with any coal stove that you have to rock the grates back and forth to shake the ash. If you can apply enough force the grates are strong enough to crush the coal that may have come lodged in between.

Besides the short strokes that was suggested above, you should also have a poker handy to dislodge any coal that may get stuck.

Another tool to have on hand is a bar that you can use to lift the grates by levering them from the bottom, as you try to move the grates. As we all do get coal or clinkers stuck in the grates. Those short choppy strokes could be spoken of as rattling the handle so that the coal bed vibrates. If you watch the coal bed as you move the shaker handle, you will see the bed seem to move up and down. Slow and deliberate works, and it is not necessary to get every smidgen of ash out, I try to stop just before I could see red coals fall.

I recently purchased a used Harman Mark III Magnafire coal stove. I replaced a Kodiak Wood stove with it. The problem is I start a coal fire using chestnut hard coal. It burns hot as heck for two days then dies out. I have to flu temp up to 400 degress and the fire is so hot, I have to wear gloves to shake down the ashes. The next morning after loading it full at night with a nice hot fire the darn thing is out. Any suggestions?