"The strong man is carrying more than a thousand plates."
Translation:De sterke man draagt meer dan duizend borden.
German is the same way, and you might be surprised at how rarely you have to make this distinction. If you absolutely need to, however, I would go the German route and specify how it's being worn or carried. 99.99% of the time, no one will misunderstand you.
Imagine if English only had wearing for both words. If I tell you, "He's wearing the hat in his hand," you'd understand that I must mean he's carrying it, because you can't "wear" something in your hand. You could probably construct a situation in which it's ambiguous and confusing, but those are rare. If one happened to occur, you could just add extra information to clarify.