Both sentences you presented read as poetic. 'Breathing the {adjective phrase} air' is a common pattern of expression in English language poetry, as is using 'air' to describe a smell.
In the first sentence, 'freshly cut grass scent' sounds odd, it would be more typical to say ‘Breathing the dewy air of flowers and the scent of freshly cut grass.’
In terms of 'dewy' versus 'moist', 'dewy' is a much rarer word, essentially only used in poetry or other 'artful' language. You wouldn't use it in day-to-day speech. Moist is a more common word. Dewy has a positive connotation of fresh. Moist has a neutral-to-negative connotation of damp or slimy (like moss or mold), though this might vary depending on dialect.
So I guess to summarize:
If you want to sound poetic, use either
Breathing the dewy air of flowers and the scent of freshly cut grass
or
Breathing the moist scent of flowers and freshly cut grass
and if you want to use a phrase people would use in casual speech, try
Smelling the wet smell of flowers and fresh-cut grass.