One of my tenants didn't pay rent on time so I went to get in touch with her and her phone is off.  All payments before were on time. I sent someone to serve the 3 day notice and he let me know that the unit looked vacant. So I go check the unit out and sure enough it was empty, the tenant even cleaned the house really well before leaving. The tenants in the neighboring units haven't seen them in a couple weeks. She did not leave the key behind.  About 8 months remaining on the lease.

My lease does not cover abandonment which I now realize should be included. So now that I've served the 3 day "pay or quit" notice properly can I proceed as if she took the "quit" option and legally change the locks and re-rent the house? Or do I need to proceed with the full eviction process even though this tenant is long gone? 

I have a deposit equal to two months rent.

This is in upstate NY. 

My magistrate has always told me that if unit is vacated and left than $500 worth of stuff is in house I could take posession.

You could call to confirm.

Then I would change locks, do any necessary maintenance and start advertising for rent.

I would change the locks if you didn’t get the keys back. Cheap peace of mind. Re-rent asap if it was me 

Personal opinion: document the condition with pictures and/or video, add a note to their file, change the locks, and move on. Apply the deposit towards unpaid rent, utilities, and other charges. If there is a refund, deal with it according to your state law.

The odds of someone vacating a home and then returning to sue you are astronomically low and they would be unlikely to gain anything from it.

Not sure what the abandonment law is in NY but in AZ we have to post an abandonment notice for 5 days before taking over possession but with all the personal property gone and cleaned, I'd take my chances and proceed like @Nathan G. suggested.