Thanks for your suggestions.
The router is getting a public IP -- same one the old router did, of course -- because the modem is set to "IP Passthrough" for the device (the router). Without that setting on the modem, the router gets 192.168.1.97. (and virtual server functions do not work).
The modem/service is Bellsouth/AT&T FastAccess DSL. The modem essentially wants to act as a router when it doesn't really have to, and you have to use this "IP Passthrough" feature to bypass that and use your own router. This functioned perfectly under the DI-524, which is what leads me to believe that the ISP/modem is not the issue.
I believe I've done an okay workaround -- I'm just using the same network settings on the static IP option. I tried this before but it didn't work well, I believe because I was using the wrong DNS settings. It seems to be working fine now -- except that now the QoS Engine doesn't seem to be doing its job.
This router might very well be headed back to DLink.