Awfully é so um jeito de enfatizar mais, não tem grande diferença.
''She's awfully tired''=She's very tired
''We have an awfully long way to go''=We have a very long way to go''
''..But three years is an awfully long time''= But three years is a very long time
''It must be something awfully bad to them''= It must be something very bad to them''
I'm awfully busy=I'm very busy,I'm so busy.
Só pra completar, existem vários jeitos de enfatizar em inglês..
Ex> I'm so tired;I'm rather tired; I'm awfully tired;I'm quite tired; I'm pretty tired;I'm very tired;I'm really tired
Hope I've helped a little bit =)