Some grammarians take issue with the senses of awful and awfully that do not convey the etymological connection with awe. However, senses 1 and 2 of the adjective were being used in speech and casual writing by the late 18th century.
Adverbial use of awful as an intensifier began to appear in print in the early 19th century, as did the senses of awfully corresponding to senses 1 and 2 of the adjective. Awfully continues to enjoy widespread use in newspapers and the like.
The adverb awful, on the other hand, is largely restricted to informal prose and to the spoken language it emulates and is chiefly an American use.