As an employer in NYS, to what address would I send the 1099 Copy 1 For State Tax Department. I cannot find the physical address to send it. Thank for the help in advance.

As a general rule, there is some type of "cover form" that goes with filings of stuff like that and the address would be on it. If you file tax forms regularly, you should just be receiving that stuff in the mail with all the instructions. If not, you need to be calling the state tax office to find out what you need and to get any necessary forms. Most decent size cities will probably also have a local office you can pick that stuff up at. (Just try the local phone book for that.)

Usually a Google search for "[name of state] state tax" will lead you to whatever that state calls their tax department, where you can find lots of information and a phone number somewhere. There's a list of #'s for NY here:
Telephone numbers (http://www.tax.state.ny.us/nyshome/telephone.htm)

This is the kind of thing you need to think about ahead of time, make the calls, find out everything you'll need to do, get registered, get forms, etc. before it's the last minute. So if you haven't already, do all that while you're at it.