I think it was you who asked about differences betwen 'nosić' and 'mieć na sobie'? So here actually 'I am wearing' is acceptable, tops (I wouldn't accept it personally but we didn't teach 'mieć na sobie'). The default sentence will now be "I wear".
And while some people would say that "I am wearing" is fine, for sure "I am carrying" is not. I deleted it from the accepted answers.
So to have it in one place:
to wear (generally) = nosić
to be wearing (right now) = mieć na sobie, although we still accept 'nosić' here
to carry (generally) = nosić
to be carrying (right now) = nieść
What exactly is an "overcoat"? А redingote? In Bulgarian the word that is cognate with Polish płaszcz, плащеница, means something like a tunic or even а shroud, which I doubt is what the Polish word refers to (unless the speaker is actually John Rambo and he has gone again on a vengeance streak in the forest).