In some contexts, "to be careful" can be used to mean "I have an ongoing habit of making sure I do [x]". This is the case in your hand-washing example; "I'm very careful about washing my hands before eating" means "I always make sure to wash my hands before I eat".
It doesn't work quite the same way in a command, though. If you're telling someone to be careful about/while doing something, you're cautioning them. For example, "Be careful when you cross the street!" is said to make sure someone uses caution and isn't hit by a car, not to tell them to be sure to cross the street.
So, in your example sentence:
"Be careful of wearing a coat."
This doesn't mean "Be sure to wear a coat", but actually the opposite; it's a caution against wearing a coat. It would be understood as:
"Take caution when you wear a coat! When you wear coats bad things can happen, and you need to watch out."
Which is probably not what you mean to imply!