Your cluster will be down if your DC will be down as you are having 3 votes in the DC.
For Log shipping you need to manually bring the secondary db online to be accessed by the applications for read write connections.
But should I use Log Shipping Secondary or Alwayson AG third node as DR, as my PDC will be down completely and I need to run application on DR databases
You can use the AG for DR - that is what it is designed for...but to do that you need to be able to move the file share witness to the DR site to allow for a quorum.
You would normally setup the cluster with 3 votes for quorum - the primary and secondary nodes in the primary DC and the file share. The DR node will normally have a node weight of 0 so it doesn't participate in quorum to determine cluster health.
For a failover - you need to move the file share witness to the DR DC and change the node weight on the DR secondary and manually migrate the listener to the DR secondary.