sick of vs sick with

What is the difference between sick of and sick with? Does the phrase ‘sick with’ sound weird to an English speaker?

I am sick of = I am tired of
I am sick of the sight of him.
I am sick of the colour of these walls, we need to decorate.

I am sick with = concerned with physical illness or ‘feelings’.
I am sick with bronchitis
I am sick with worry.

(No, it doesn’t sound weird.)