Quite frankly, if you are dizzy, I would not want you to be transporting my children. I know it sucks to hear that, but nobody should be driving anyone, in any vehicle when they are dizzy. It's just not safe.

That said, perhaps you should seek a second opinion about your health issues/lack of sleep/shakiness. They might have some other thoughts or ideas, or be able to help you get more time off till the problem is taken care of.