I recently married. My husband owns a house in Virginia. I want to know if my name can be added to the deed, or title and not to the mortgage

Sure, go a local title company and tell them you'd like to do this. It should be fairly inexpensive and then you'll know it's filed properly, etc. (Probably something like a "Quit Claim Deed" is what you'd do.)

That said, technically the lender could call the note due if there's a change to the title. Almost all mortgages have a "due on sale" clause. It basically means that if the house sells / changes title / etc that they can immediately call the note due in full. I've never heard of this happening on a performing note (meaning payments are current), but in theory it's a possibility.