Learn English – Be sure to [Verb] Vs Make sure to [Verb]

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!

You can use either "to make sure" or "to be sure" in your sentence presented, without any difference in meaning. However, according to Ngram, the former is becoming more common in the last couple of decades.