Update: With the way I had it set up, I was getting an alarm every few hours that one of the hosts lost connectivity to the SAN. It would regain connectivity after a few seconds. Sometimes it was one host, sometimes the other, but not both at the same time.
So I tried again to set up a VMKernel on a separate vSwitch, and I figured out why it wouldn't work before: I tried not specifying a VLAN in ESX, since the physical NIC was plugged into a port on the physical switch that was already set to be on the VLAN that the SAN is on. It works that way and sees the SAN, but still loses connectivity like before.
I found and looked through VMware's "iSCSI SAN Configuration Guide", but didn't find anything about setting it up for multiple hosts. Am I supposed to make another iSCSI Initiator on the SAN for the second host to connect to? I have them both connected to the same one with all the same settings, including the same IQN. Is that how it is supposed to be?