Hi Peter,
You're correct that you should ideally have one GTM per data center. If the GTM fails or critical resources in a datacenter fail the affected WideIPs will be resolved to a data center that has the virtual servers available. This is DNS based load balancing so there isn't really a concept of active connections. Depending on the WideIP load balancing algorithm you can enable persistence and mirroring of the persistence records.
If you want more specific recommendations I recommend you contact an F5 or partner SE to discuss your exact requirements.
Aaron