But no, you're probably not responsible for the debt and should be able to walk away from the property and allow the inevitable foreclosure to show up on the estate's record. However, only a good local real estate attorney, probate attorney or other qualified professional can say for sure, plus formally advise you how to best proceed by examining all details. Of course, laws do differ from state to state, sometimes significantly, so do your research carefully. There may also be near-term deadlines for actions you need to meet to further protect yourself.
You don't mention any children who are heirs, so that's apparently not a factor, nor do you note whether a legal will was drawn up by your husband. Regardless, I can assume the estate did not go through the probate process, where outstanding debts get settled out of the estate's money (and the house sometimes gets retitled in name of the spouse), because the mortgage company obviously hasn't been notified of your husband's death.
However, if you did inherit any money from your spouse, you should know that those funds might be subject to claims from the mortgage lender and other creditors to settle outstanding debts on the estate when they do find out. In some cases, lenders have been known to claim that a surviving spouse acted fraudulently by taking inheritance money before the estate was settled. Hopefully this isn't an issue for you. However, if you were a beneficiary of a life insurance policy, that would not be considered part of the estate. One thing you may be held responsible for, though, are current property taxes, since you continue to dwell in the house as a "person in possession."
To some small degree, I understand your reluctance to notify the mortgage company. But hopefully, you or other kin have notified your husband's insurers, credit card companies and other creditors of his death. Those delays could harm you instead of help you.
Typically though, if you're not on the mortgage, you're not personally liable for that debt. Moreover, your imminent departure shouldn't adversely affect your credit, though you may want to keep an eye on your credit reports anyway in the coming months just to be sure.
Good luck in finding a new place!