No, the switch should not make any difference.  That should work fine.

If devices that are connected to your router are getting DNS addresses that are the same as your router, that is correct.  Most home routers act as a DNS server, so they hand out their own address as the DNS server.  The router then forwards the DNS requests to the upstream DNS server.

Check your subnetting on the 2Wire RG and on your router carefully.  By default, the U-Verse RG's DHCP range is set to 192.168.1.x.  This is the same DHCP range that many home routers hand out.  If the two DNS ranges are the same, Internet will not work.  You must change the LAN addressing of your router to 192.168.2.x (router address: 192.168.2.1), or you must change the 2Wire DHCP range on this page:

http://192.168.1.254/xslt?PAGE=C_2_3

Once the two routers are using different LAN subnets (i.e. different router addresses and different DHCP ranges), you should have Internet.