You are right to not want to run multiple DHCP servers on the same network, that can end up giving very random results.
Right now it sounds like you have both wireless routers plugged into your main router by their internal ports, using them basically as access points instead of routers (just getting the wireless users connected into the main network and letting the main router do the real thinking).
If the wireless router the students are on allows you to setup your port blocking rules to specific addresses, you may be able to on that one set it to block all communication, then allow specific rules to your internal servers. Then connect that router to your main network by the Internet/WAN port on the router and enable DHCP on it. That will make it so that the students are segmented on their own network and can only access your main network routing through the wireless router, which has the rules setup on it.
So, you would have:
Internet Connection <-> Main Router WAN Port
Main Router Internal Port <-> Teacher Wireless Router Internal Port
Main Router Internal Port <-> Student Wireless Router WAN Port
Anyone connected to the main router or the teacher wireless will be in one network and have normal access. Anyone connected to the student wireless will be in another network behind the student wireless router and be subjected to the rules setup on that router.
Otherwise, you'd want to look into setting up a DMZ to separate the students from everyone else, then setup your specific rules they are allowed to do. Your current equipment may not support this, your main wired router would be the one to look at.
Stick to your modem.
Then get an Mikrotik 750 o 450 or one of their G counterparts to do the routing for you. The modem stays working as modem only.
Then natuarllly you betterk now what you are doing - that is higher end pro stuff.
Best Answer
Keep in mind that the 2Wire Gateway device installed with your U-Verse service is more than a "modem", it is a router, DHCP server, and WiFi Access Point as well.
So, connecting another similar device to it can introduce conflicts, in routing, in IP address allocation, and so on.
Right off the bat, I'm thinking you may have overlapping DHCP scopes, because 'out of the box' most home/small biz routers sold allocate addresses in the 192.168.x.x range (private IP's). This is the range that the 2Wire's DHCP server uses. So what happens is, you end up allocating the same IP to two separate devices on the same network - when this happens, BOTH see the conflict and stop talking. Your release/renew exercise is a stopgap (essentially you're getting a "new" IP) but depending on which DHCP device you get it from, you may end up with the same problem again.
To fix the issue, the easiest way to do this would be to configure the second router to "bridge mode", essentially turning it into a switch with a DHCP proxy. Rather than make routing and address allocation decisions by itself, it would then send those requests back to the 2Wire unit. Or, ditch the router completely and install a switch in its place.
If you must have the second router on your network, configure it to use a completely separate range (DHCP Scope) of IP's, and then set it to route THAT subnet to the 192.168.x.x subnet managed by the 2Wire gateway. You'll end up configuring a lot of extra options, but once it's set up, you shouldn't see these issues continue. Be sure and save all of your settings in a text file, Excel file, or database, on a flash drive or on another system, so that if you ever have to replace either the 2Wire OR the second router, you don't have to redo all of the research, and can get straight to configuring things to work well together.