I have a set of ASA that has a DMZ switch which works great on production network. I have another set of PIX that is used for my backup Internet line, when I plug them into the same switch as the ASA the DMZ servers are unavailable. I am using OSPF on both the ASA and PIX. They all have different DMZ addresses on the interfaces. Why is the DMZ network going down when I plug in the PIX for the backup Internet connection?
Without knowing more about the network topology, I can't say for sure.
Why are you running ospf on your dmz interface? Are there more networks off of the dmz besides 192.168.0.0? Do you intend to use it as a transport network between the firewalls? If not, I would turn ospf off on those interfaces and just advertise the connected network.
You probably have to increase the cost on the inside interface to make sure the pix is only used in case of failure. Check what cost your asa is advertising and go above that.