hm...I still don't think it's necessary for DD to trust Snape...at least not at the time he changed sides. Just imagine:
You are fighting a war against the most powerful evil wizard that ever existed, and you're supposed to be the only one who is possibly able to stop him. You try your best, but nevertheless, he manages to gain more and more power, innocent people are tortured and killed, families are torn apart, and there is no more safety, nowhere. Then, at the climax of his power, one of his highest ranking followers offers to change sides and work for you as a spy...what would you do? Turn down such an offer? Even if you didn't fully trust him, wouldn't you run the risk? I would...
But I really like the idea of Snape saving DD's life