Ok, perhaps we need to take this back to basics.
The fact your computer is currently connected to the Uverse modem+router is completely irrelevant. Nothing says you can’t DISCONNECT your computer from that device! That’s what I’m trying to get across to you.
Put the Uverse modem+router COMPLETELY out of your mind. In fact, if it helps, turn it OFF. Disconnect the ethernet cable (or wireless connection) you have to it from the computer. Now you have nothing but a computer standing there connected to NOTHING.
Now take one of those other modem+routers and connect it to the computer over an ethernet cable and reboot the computer. It will connect to the device and receive a TCP/IP configuration on ***that*** device’s network, whatever that may be, because it will use the DHCP server it offers to configure your computer, automatically. No, it won’t be able to get you connected to the internet. But all you’re trying to do is get connected to that device and reconfigure it (which will include giving it an IP address on your new Uverse modem+router’s network and disabling its DHCP server).
Now do the same thing w/ the second modem+router. Connect it to your computer, reboot the computer, and reconfigure it.
Finally, connect your computer back to the Uverse, reboot the computer, and patch the modem+routers you just reconfigured to the Uverse as well. If done correctly, you should be able to access all the devices and have internet access again.
You’re making this way too difficult for yourself. You keep thinking (erroneously) that somehow, for some reason, you need to stay connected to your new Uverse modem+router to accomplish this task, which is NOT the case. You’re over-thinking the problem and just getting confused in the process.