I wonder if Google will see my new site as duplicate content to old site that was already down for 3 months and the site was now out of Google index.
hmmm...yes its true, your site is out of google index because of a trigger and if you placing its content to new site, it may also repeat it self.
If its not indexed then no. you should take the old site offline or do a 301 redirect for better results.
Depending upon the cached content it may consider your site as site having duplicate content. Google keeps on refreshing and getting new cache , 3 months is the usual time ones new site gets cached . Most probably Google may not consider your site as duplicate . Canonical attribute will work great if you have the old site under your command.