This all depends on which state the property is situated.  Differing state laws have varied requirements.  In some states, there is a homestead right that the non-borrowing spouse has.  So, even though the non-borrowing spouse is not on the promissory note, he or she must sign the mortgage or deed of trust which subordinates any homestead rights to the rights of the mortgagee [the lender]. 

There are no simple or easy solutions if you are in a state which requires that signature.  Divorce is one option.  Finding your husband and having him either quit claim the deed [which may or may not remove the need for his signature on the security instrument] or sign the mortgage or deed of trust is another. 

Your best option right now would be to consult with an attorney in the state where the property is in order to find out for sure which avenues are open to you.  It is possible that you are working with a lender unfamiliar with state statutes.

Based on your question, the issue could be who is on the title to the home, NOT who is on the loan.  If he's only on the loan (very rare), and you qualify on your own, you should be able to proceed.  If he is on the title/deed to the house, you've got issues.....and potentially very EXPENSIVE issues.

I've helped people in similar situations and they were held up by their "ex" after one of the "ex's" friends suggested that they were entitled to a good chunk of the equity in the property.  This is a serious issue, and I would strongly recommend legal counsel, just like the other poster.

In most of the situations I've seen, the "ex" was paid something through the loan closing to induce them to remove their legal rights to the property. 

Probably not the answer you wanted to hear, but very likely the correct one.