What is the ruling on working in banks and usurious institutions, knowing that my experience is limited to this field and I cannot find any other work to provide for my family's living requirements?

You have done well in seeking out what is lawful. Usury (riba) is among the most severe sins, and its sin is not limited to the one who consumes it, but also includes those who assist in it. Praise Allah for abandoning the unlawful and strive to seek out what is lawful. If you can earn a living through a permissible means, then it is forbidden for you to return to the forbidden job. However, if you cannot find another means and your remaining in your current situation would lead to your destruction, or the destruction of your family, or harm befalling them, then you are in a state of necessity regarding this work. Nevertheless, necessity is measured by its extent, so when the necessity ceases, you must leave it.