OK, English dudes and dudettes. To me, when you use badly it means poorly. For example “I felt badly about that” means your hand arent working too well. So naturally I read the above sentence as the person wants have their name in the subject of an email, but wants to do a poor job of it.
“To want badly” is moving towards idiomatic status – if St. Paul were writing today, he’d discuss his efforts towards perfection with “I want to be as good as possible badly!” Also note “a badly needed improvement” where “badly” seems to function purely as synonymous with “very much,” “greatly.”
That specialized intensifier meaning to one side, let me take note that a definition that makes a word synonymous with another may not always be the ideal way to explain the word. “Badly” is an adverb that places a modification on the word, usually a verb, to which it is in grammatic affiliation. The classic explanations of predicate adjectives after verbs other than “to be” get into this – If the dog smells bad, it needs a bath; if it smells badly, it needs a veterinarian’s services to cure its rhinovirus infestion.
Churchill was complaining about the rule against ending a sentince with a preposition, not a dangling participle. He correctly was indicating that it is perfectly good English that a preposition was a perfectly good word to end a sentence with.
A dangling participle would be a construction like this:
Walking down the street, the tree looked beautiful.
The above sentence implies the tree was walking.
Though there are grammatical (“hand arent”) and punctuation errors (“arent”) in the OP, there is no dangling participle.