Report all your unemployment on your federal return.
You will be subject to NY tax on any unemployment received while still in living NY (in 2020) - depending on how much that might be, you may have to file a 2020 NY return.
You will be subject to CO tax on any unemployment received after becoming a CO resident. As a CO resident on Dec 31, 2020, you will probably have to fie a CO part-year resident return (will depend on your total and types of income)
Taxes are based on the state where you lived at the time you received the payments not where the company or organization is located that distributes the payments.
