We are looking for a way to manage tickets that come in after hours or weekends.

I wanted to see what others are doing to notify on call staff of tickets.

One of my requirements is that it does not alert everyone, just the on call team, and that the alert is only generated off hours. I.E - I don't want to get an alert every time a ticket is created M-F between 8am -6pm.

There's a ticket filter you can use with Dispatch'r that lets you choose from Business Hours, Non-Business Hours, Holidays.  You can use that to catch the tickets and notify or alert whomever you would like.