My first and last observation is you consult with an attorney in your state who has experience in probate law. There may be significant issues relating to the mortgage, your spouse's will (if any), your state's laws, and even life insurance that you did not mention in your question that have a significant impact on your rights. An attorney will review all of the documents relating to the property and give you advice that is accurate for your situation.
Generally speaking, how the property is titled controls who takes the property upon the owner's death. I will not try to summarize two semesters of property law here, but if your name is on the title to the property that is good news. However, if your name is not, that is not necessarily bad news. This is one of the first things your attorney will look to.
A will or trust controls how a person can control how their estate is settled. (An "estate" is all of the assets owned by the descendant.) If your spouse had the property in his name only, then the probate court will look to any will or trust for direction. If the will or trust directs that the property be given to you, then the court will do so (assuming, of course, that the will or trust are valid). If your spouse had a will, he may have given directions for handling the mortgage in the will. There may be an insurance policy for this purpose.
Finally, if there is no will or trust, the court will follow its state statutes and common law for settling the estate. These vary, but generally speaking a spouse will receive half, and surviving children split the other half. However, I must remind you that each state has its own rules of intestate succession, and your attorney will advise you on your rights if your spouse died without a will.
Regarding the mortgages (the home equity line of credit is a form of mortgage), a mortgage is an encumbrance on the property. Encumbrances stay with the property as it changes ownership, and remain until satisfied.
One reason I spent so much time discussing the possible paths to your being on title of the property is to give you an idea how long it may take for the probate process to complete. The executor (if there is a will) or administrator (if there is no will) is responsible for carrying out the tasks in the will, and managing the financial affairs of the estate during this process. It is the responsibility of the executor or administrator to pay the mortgage during the probate process. Ultimately, the executor and administrator work for the court and are employed until all claims are resolved and all property is distributed.
This brings me back to the beginning of my answer: Hire an attorney to help you understand the process and guide you through probate.
I hope this information helps you Find. Learn & Save.