*Gary T
“I feel badly about forgetting my aunt’s birthday.”
In this case the adverb “badly” is attempting to modify the pronoun “I.” It is not correct.*
No case for arguing a parallel to constructions such as “I itch badly”? I agree with you in general that it is at least contentious - this is just throwing an idea about - but I don’t think it can be taken as automatic that the “badly” modifies the “I”.