When I separated from XAH, I knew that it wasn't a "trial separation", but I still took care of the financial aspect of things myself. Even though I had a lawyer, she only got involved after mediation failed.
Since we were both on the lease for the apartment, XAH agreed to take over the lease. We had to serve the landlord via certified mail with the paperwork attesting to his willingness to take over the lease. Once she accepted, I was liberated from any and all obligation to pay whatever rent XAH would default on (which was smart of me since he defaulted on the rent 3 months after I left and moved out in the middle of the night).
I transfered all utilities in the apartment to him and separated our bank accounts, removing his access to my money.
In your case, if your H is on the mortgage and he cannot afford to pay it, then he needs to put it on the market. This is where a consultation with a lawyer would come in handy. At least go for 1 consultation so that you can get a clear picture of your financial and legal obligations as per your geographic area. The laws vary greatly from state to state. In some states, you could be held liable for half his debts (mortgage included) if you were to separate...