Why is the cable modem's DHCP server running when it's supposedly disabled?
You'll need to contact the vendor on that point. Maybe there's a bug in the firmware you're running; maybe the config displays but hasn't been applied; maybe there's a second step to fully disable it.
Why are addresses in the 192.168.0.x range "bad" on our network?
You'll need to do some reading on IP basics. Those addresses could be made to work if you either adjust the subnet mask of both subnets to be larger (so that they're the same subnet), and/or set up proper routing between subnets. That's probably not what you want though; it's not what I would do.
What can I do about it?
As I said above - contact the modem vendor about the apparent bug.
It's also worth posting your cable modem's make and model, in case someone here's familiar with it, plus a network diagram. Typically, if you're using the cable modem strictly as a modem and have your own router, the cable modem's LAN interface shouldn't be able to receive DHCP request packets from your office LAN, it should be in a separate L2 domain.
You may want to consider hiring a network pro for a 1-day engagement to review and fix your setup.