Yes they are different, although the definition and distinction my only be of interest to scholars of classical rhetoric. There are, according to this article from the University of Alberta, three figures: Figures of Word, Tropes (Figures of Speech), and Figures of Thought. All three are tools in rhetoric for conveying information.
Figures of word involve the actual construction of a sentence, including word selection, placement, and repetition.
Figures of speech are when words or phrases are used in a sense other than their literal meaning for dramatic effect. Hyperbole is a figure of speech.
Figures of thought are the styles that rhetoric can take, the way an argument is approached. A simile is a figure of thought.
They are different concepts, but not mutually exclusive. Examples of figures of thought could be figures of speech, and contain figures of word. Figure of speech is, by a wide margin, the concept that we are most familiar with. And we are familiar with specific types of all of them, including, hyperbole, paralipsis, metaphor, and allegory, but their technical classification wouldn't be of much interest to anyone outside of an academic setting.
They're puns that depend on the use-mention distinction, as many puns do, so we could reasonably call them "use-mention puns", a phrase that's had some use, but not a lot.
Best Answer
I don't think that there's a name for this figure of speech—each phrase uses an adverb in an accepted form. For example, take the definition of "awfully":
The phrase "awfully nice" uses the first meaning of awfully—"very; extremely". I think that the phrases you quoted are perfectly good English; they don't employ a figure of speech.
However, one might be able to call them oxymorons, but they will not be fully accepted as such. From this list of figures of speech:
One example as given is awfully nice and awfully good.