Network appliance having forwarding traffic mode to reduce traffic loss and related methods

A network appliance is provided for establishing sessions between client devices and a network server(s) for exchanging network traffic therebetween. The network appliance may include a memory and a processor cooperating with the memory, with the processor being operable in a normal traffic mode and a forwarding traffic mode. The processor may be configured to establish new sessions for network traffic based upon new session requests from the client devices, and forward network traffic associated with prior existing sessions from the client devices to the network server(s). When in the forwarding traffic mode, the processor may forward network traffic not associated with a prior existing session or a new session request to the network server(s). When in the normal traffic mode, the processor may block network traffic not associated with a prior existing session or a new session request from reaching the network server(s).