How can I get the sharpest handheld shots at maximum zoom of a moving subject?

There are several important things. I leave the most important to the last.

1. Use the widest aperture possible. If needed get a better lens. Of course, you can not do that in your camera, so it is a limitation of your camera.

Force aperture to the lowest possible value to give the highest depth of field in case focus is not so accurate

2. No. If you want good focus you need a lens and camera that give you good focus. You can not sacrifice something that will give you real problems, like lowering the amount of light at the expense of something "just in case".

Force shutter speed to a very high value

3. You set the speed to the highest value possible. Given your aperture and ISO. I will address the "force" word for later.

Force ISO to a low value

4. Nop. Use the ISO to the highest value you are willing to use.

You need to test your camera. Change the ISO and see the images on a computer screen. If the images at some High ISO value, for example, 6400 are too noisy that you can not live with them... do not use that value.

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Now the word "force".

If your camera is limited, as all cameras are, you need to live on what you have.

5. A way to force the shoot to have less motion blur is to underexpose it.

If a happy image is shot at 1/250 but the image is still blurry by motion blur, then underexpose at 1/500. Then you compensate it on post. The result is similar to use a higher ISO, this is... more noise.

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6. Get a good tripod, or way to stabilize your camera. It can be a rock, the car, the floor, a wall.

7. And finally... In some cases, you can reduce the shakiness of the camera itself using a timer or remote controller so you do not touch the camera, and leave it alone for some seconds so the shakiness of your hand dissipates.

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And the most important factor is...

8. Light up the moving subject!

If it is a football player at night... ask the stadium to turn on the lights.