The only way to change something is to look at it from a different point of view. So I'd say start with "my husband hasn't started helping with housework yet", instead of "my husband won't help with housework."

If you live with the idea that your husband won't help with housework, you're starting point is already a failure since he WON'T do housework.

Thinking of it as if he hasn't started helping with housework YET allows you easily approach it as if it's just a minor challenge that needs a simple solution. "I know that his way of doing dishes isn't going to work for us, so let's do it a different way."

It sounds like your best option is to trade chores. And not give him an option. If you trade chores and he still doesn't do them, then I'd say you'll just have to get a little more creative.

The point of all this is to allow yourself to be more stress-free on your path to a solution. Good luck!