My mother and father were going through a divorce, but it wasn't final at the time of her death. She had made me her sole beneficiary. Am I responsible for any of the debt my parents had?
No you are not.
Sorry about the goings on in your life that is certainly a lot of drama in addition to the loss of your mother.
An advantage of having assets pass without going through probate is they do not have to stand for debts of the estate (usually). The life insurance is yours free and clear.
You do not need to pay for any final expenses, but I would suggest you do. Make sure your mom has a nice funeral.
Presumably your mother was going to receive part of the marital debts if the divorce had gone through. Is your father now stuck with all debt or does your mother's estate get her share? I don't know. You may want to talk to your mom's lawyer about that.
Let us assume that your father now has all the marital debt. Should you pay some of it? That is a judgement call depending upon your relationship with your father, his relationship with money, and the reasons for the divorce. But that is a judgement call and may alienate you from your father for some period of time.
If her estate gets her part of the marital debt, then there is no need to pay them.
You cannot inherit debts, and the only cases where life insurance can be used to pay the debts of the estate of the deceased are when a living beneficiary was not named on the policy.
It's possible for the life insurance beneficiary designation to be challenged in court, but that is much harder than challenging a will. Unless your father manages a successful legal challenge to you being named as the beneficiary, that money is yours and does not include any obligation to pay debts incurred by your mother.
There is one example I can think of in which you would be responsible for the debt of a life insurance policy owner -- if your mother had universal life insurance or whole life insurance, she would have had the right to borrow from the issuer against the cash balance of the policy. The balance of the loan would be deducted from the death benefit when the policy pays out, effectively forcing the beneficiary to pay her debt.
This debt could never be larger than the cash value of the policy and, of course, if your mother had only other kinds of debt, this would not apply at all.