Can a person that lived and worked in two states in one year draw unemployment from one state, then when the funds run out, draw from the second state?

This is REALLY a legal question, NOT a tax question, but I will answer to the best of my understanding.

You can collect from the second state only if you do not "double dip", i.e. collect benefits for the same period.

The administrators of the unemployment compensation funds (usually the state's Department of Labor) DO coordinate with each other to prevent this from happening, but, if you apply and get double benefits, and then the administrators discover the error, they will pursue you in court to recover the over-payment.

Further, if they suspect criminal intent, you may get prosecuted for fraud.