Duplicate IP after L2 extension across Datacenters

You can't have both data centers sharing layer 2 and not expect some problems if the boxes are identical. You need to go to Local Traffic, Virtual Addresses on the down system and disable the Virtual Address. You may need to disable ICMP and ARP as well. This won't help you with Self IPs. Your only option there is to have different IP addresses, have the layer 2 networks isolated, or to have the units setup in an HA Pair.

The upside to adding the DR site as a Sync Failover peer is that in the even that the DC fails, the DR site should be able to go active automatically without having to turn addresses on and off. The downside is that 1) depending on latency you may have active active scenarios and traffic disruption and 2) you may find that your DC has failed over to your DR site when you were not expecting it. This could also be a service disruption.

Probably the best option would be to isolate the l2 networks and use something like BGP to handle failover scenarios, but you would need to chat with your network team to make that determination.