I just recently moved how can I be sure the unemployment has my correct address to send my W2 forms and when can I expect them.

I forget the form number now, I am sure the right person here will have it, but it is not a W2 you get from unemployment, it has a different form number.

In every state I know of, you will get it off the internet, go to the page where you were doing your weekly filing and it should tell you how to first give them a change of address notice and second how to downloard your form for taxes

The Form is Form 1099-G.

In most state, you can download YOUR Form 1099-G direct from the state's Department of Labor website.