Alright, the new cable modem from my ISP showed up. Now before I connected everything I unplugged everything from my wireless router, disabled the wifi and connected my laptop to 1 of the LAN ports. I was assigned the proper 192.168.1.xxx IP address automatically. I ran the rogue dhcp server software and it only shows 1 dhcp server, 192.168.1.1 which is my wireless router.
I plugged in my new cable modem, let it load, plugged it into the WAN port on my wireless router, connected my laptop and was assigned a 192.168.100.xxx IP address. I ran the rouge dhcp software and it's showing 2 dhcp servers, 192.168.1.1 and 192.168.100.1. When I point my browser to 192.168.100.1 it takes me to the admin page for my new cable modem, the Cisco dpc3825.
The Cisco dpc3825 is a wireless modem that my ISP has put in bridge mode. I tried to login to the modem but all my login attempts are rejected. After doing some reading online I found that if the modem is bridged you aren't able to log in. The modem is also not broadcasting a wifi signal which leads me to believe that it is in fact in bridge mode.
I guess we've determined what is causing the problem, now I'm wondering how I go about fixing it? Looking at some screenshots of the Cisco modem admin pages, it looks like it has most of the features I need to setup my network properly so I could have it put back into normal mode and use it as a modem and wireless router. Although there are not a lot of positive reviews out there for the unit as a router, most people suggest getting it bridged and using your own wireless router.