Is it permissible for the one in charge of a third of my deceased grandfathers' wealth to pay my accumulated debts so that I can provide for my family under my current circumstances?

If your salary does not meet your essential living needs and you have debts you cannot repay, then you are among the gharimin (those in debt) who are eligible for zakat (definition), even if you incurred the debt due to a sin from which you have repented. Therefore, you are more deserving of having your debts paid from charitable funds designated for that purpose.