May I deed property under mortgage to my sons while retaining responsibility for paying same?

Yes, but I would never keep the mortgage in my name. You are responsible for payment, but do not have a any legal rights to it. What happens if he dies and leaves the house to someone else? You still pay. Check with the Mortgage company to get their permission, before you do anything. They may accelerate the mortgage to be paid off. In short - do not do it.

Leaving aside the practical question of whether this is a smart thing to do, the legal answer is that it depends on what your Security Deed or Deed to Secure Debt says. In most standard documents, any transfer of ownership interest is an event of default that would allow your Lender, usually at their option, to declare the full amount due and to foreclose if you don't pay. So, read your security instrument, and to be extra safe, get written permission from your lender to do this before you do it. However, even before that, you may want to discuss the wisdom of such a move with an estate planning attorney. There may be better options.