'Fall to...', In the sense of 'began to', was often used in the past in situations such as "The hungry travellers fell to eating" or "The men fell to arguing over the spoils."
The writer, George Eliot, certainly used the phrase 'fall to sleep' in a publication of the 1870s, but that construction is rather archaic now.
'I felt asleep' is along the same lines as writing 'could of' because it sounds like 'could've' (a contraction of 'could have') when spoken. Avoid using both.
glad you mentioned that one Tim, that's another that really bugs me!! I haev even heard people on television say 'Could of.........' Aaaaaarrgh!
Thanks for all the replies