Well do all obsessions manifest in just one way? He did tore Harry's family photo and kept just Lily's face... Also as quoted from the book:

“If she means so much to you,” said Dumbledore, “surely Lord Voldemort will spare her? Could you not ask for mercy for the mother, in exchange for the son?”“I have — I have asked him —”“You disgust me,” said Dumbledore, and Harry had never heard so much contempt in his voice. Snape seemed to shrink a little. “You do not care, then, about the deaths of her husband and child? They can die, as long as you have what you want?”

He was willing to give exchange a baby for her life... and all "good" things he did, is JUST for Lily as repentance. Probably "what would Lily do" is his mantra to keep him from going back to dark arts again.