"so badly" as an adverb meaning "severely, very much" seems to only modify adjectives when they are related to injury. It doesn't work with "tired." I can't think of non-injury related adjectives that it works with.
Examples where it works:
He was badly hurt. He is injured so badly. She avoided using her badly bruised foot. We took the badly damaged car for repairs.
PS. I'm sure you know "badly" is also used as an adverb meaning "very much" to modify verbs related to desire.
Eg. I wanted it so badly. She craved chocolate so badly. I badly desired some rest.
@QLiu I understand your confusion. I also can't explain when or why "badly" can be used as an amplifying adverb.
"I'm struggling badly", "the room was badly cluttered", "she was badly frightened by the dog", "I miss you so bad"... these all sound fine to me. But "badly tired" sounds wrong. I don't know why, and I can't find any explanation online aside from the rule about injury-related adjectives and desire/need-related verbs.
I guess it is simply a matter of collocations (words that commonly go together) in English. You have to just learn how to use that phrase through exposure. 🙁