I called the Tax department for NY state and they asked me to file an amended return using form IT-203X. Only problem is that I am refiling for year 2013 and NY state website only has back to 2014 IT-203X.

However, I found the 2013 IT-203X form on another tax website.

I was wondering if I should use it?

I think its much easier because then in the parts of the return where they ask about 'Did you itemize your deductions on your 2016 federal income tax return?' or 'Did you or your spouse maintain living quarters in NYS in 2016?', it just makes it easier since it says 2013 (the actual year I am amending) and consistent. Plus the standard deduction amounts in the form are also the right ones (as opposed to the higher amount for 2014/15/16 forms).

Any reason why I shouldn't use 2013 IT-203X form I found?

use the Form IT-203X that you found. The odds are the NY state tax department will accept it as a valid substitute form.

Do you think I should put my current address (not in state) or use the address I had initially used in 2013 for my tax returns?

Use your current address.

Otherwise, how would the NY state tax department contact you about the amendment?

And where do you think they are sending the check?