Yes, it sounds natural (but watch your punctuation!). 'I couldn't stop feeling sad' has a slightly different meaning.

It's absolutely natural (I have actually said this before) Using stop would change the meaning of the sentence. "I couldn't avoid feeling sad" means that you previously did not feel sad, but ended up feeling sad because you couldnt avoid the matter. "I couldn't stop feeling sad" means that you previously did feel sad, and continued to feel sad because you couldnt stop the feeling of sadness. Hope this helped :)