Most of Verizon's DSL areas which were DHCP, were sold to Frontier a bit ago. You're most likely using PPPoE.
Do you have another router? Bridge mode is only meant for those who wish to use another router, but want to disable the routing functions built into the modem. If you're using the D-Link as your router for Wireless and Wired access, the behavior you're seeing now with 192.168.1.x "RFC1918" addresses is normal, due to the fact that you only have one Public IP, and must use NAT to share that IP with multiple computers and devices.
You might be perhaps looking for Port Forwarding, instead?