Do US state-level governments shutdown by default if the US federal government shuts down?

NOT EXACTLY. The US Government Shutdowns refer exclusively to the US Federal Budgetary obligations. In theory, the US State Governments should not be affected, since their own budgets operate based on votes and balancing on a state level. However, in the US, most state governments receive a large percentage of their funds from the federal government. As a result, a number of state government functions are frozen or have their funds reallocated to make up for the shortfall from the federal government. Additionally, many of the government employees that people are familiar with are actually federal employees, such as local branch offices of federal executive agencies, post office workers, and the people who run national parks and monuments.