Deeding the property won't do squat with regard to foreclosure. The only thing that is preventing foreclosure is the fact that the bank is still getting payments on the mortgage (and they haven't caught wind of the ersatz transfer of the property and invoked a due-on-sale clause).
The mortgage has NO relationship to the property ownership.
You can't (except in certain court cases) "get" a deed. Deeds are documents from a person with some (potential) ownership interest who is giving it to someone else.
Your father is the only one who can grant a deed (either the quit claim he already provided or a warranty deed).
If he wanted to make a warranty on the grant, he certainly could if he wanted. If he doesn't want, there's nothing you can do to compel him to do so. Just because you're voluntarily paying his loan doesn't give you any rights here.
Do you know what a "warranty deed" is compared to a "quit claim?"
Both transfer some ownership interest. A quit claim says that the seller makes NO representations as to what that interest is. I can quit claim deed my interest in Wrigley Field. You'd have a worthless piece of paper as I don't have any such interest. A warranty deed purports that the owner either asserts that he has EXACTLY the interest specified in the deed or that he personally hasn't done anything to cloud the title (like sold interests to other people or take out undisclosed loans). These are your general and special warranty deeds. It gives you the right to sue him if it turns out the interest was somehow compromised.
The problem is that with a near indigent seller, the warranty doesn't mean much. You can sue him all you like and win but he's got nothing to pay you with.
This is why you always obtain TITLE INSURANCE on a property. I wouldn't even take a lot from my sainted aunt without owner's title insurance. The title company will search the deeds and verify the ownership claims of the grantor and then provide INSURANCE that their research is correct. If it is found not to be they will have to correct it or compensate you for their error.
