Hi there,
Typically, a deceased's funeral costs come out of his estate. They are a priority obligation that must be paid before most other estate debts. If, however, there still aren't enough funds to pay for the funeral, the person who signed the funeral contract will be responsible for the outstanding amount. You can check with the executor to find out whether there are sufficient funds to cover the funeral. You can also refuse to have anything to do with arranging his funeral or burial.
Also your ex may have had a pre-paid funeral plan, life insurance on his superannuation or some death or pension scheme through his employer that may cover funeral expenses.
As regards any outstanding debt that his estate is unable to pay, short answer is that you will only be liable to cover his debts if the debt is secured against your asset, the debt is in joint names with you or you have guaranteed the debt. If none of those scenarios apply and there is not enough money in the estate after all the assets are sold, the debts may not need to be paid.