Just a suggestion:
The solution may lie in manipulating the first two words - "say this" - using "across" as a key, and also assuming that "this" means "crossword".
"Say": if we assume here that "across" means "cross out a" we are left with "sy".
"Crossword": if we assume here that "across" means "without cross" - i.e. "a" in its Greek meaning of "without" - we are left with "word".
If we put these two together we are left with an angram of "drowsy", which fits in with the sleep theme.
Anyone got any ideas for 5 across? At the moment I've got "w-y-e" but can't make it work.