If the ISP connection (AT&T) is only providing 18 Mbps, then you are going to struggle with more than one HD TV session running at once. Even at standard definition TV, at least half the ISP bandwidth will be accounted for by your 3 Hulu devices. If the security cameras are sending continuous video streams to the Internet, then that can be another sizeable chunk of bandwidth depending on the resolution and frame rate.
As others have said, devices look for WiFi networks to connect to. It is not the router looking for devices. The router simply advertises its presence.
There is no reason why a device that has previously used Verizon should not be able to connect to AT&T as an ISP.
The only real answer is to get an ISP feed that is capable of handling the peak demand on the connection. You need to determine the bandwidth required by each device and sum this up. Add at least an extra 25% to 33% as a margin and go for that.
In addition, putting a lot of devices on the same WiFi network at the same time is also a potential bandwidth bottleneck. Fixed high bandwidth devices such as TVs are normally better when they are connected to the router by Ethernet, assuming that they offer an Ethernet connection. Another possibility is to connect a wireless access point to the router and set up an additional WiFi network.
For devices closer to the router, using 5 GHz instead of 2.4 GHz is another way of reducing WiFi bottlenecks; however, the ISP bandwidth from AT&T is the first thing to investigate.
Note: The ISP connection simply provides a connection to the Internet. It does not carry Ethernet or WiFi. The ISP connection is converted to multiple Ethernet and WiFi feeds in a wireless router or gateway.
I hope this helps.