How do I split the internet connection coming from my main Uverse DVR box?

The issue that I am trying to solve is utilizing my internet connection via cabled connection (not WiFi) for multiple devices (i.e. a Smart TV and Nintendo Switch) to the main Uverse DVR box that I have setup for the main television in our den. The said main Uverse DVR cable box is receiving its signal via coaxial connection to the wall in the den. The At&t Uverse router is connected via coaxial to the wall in my office, so connecting any peripherals via a cabled connection in the den to the open ports on the router isn't possible because it is in an entirely different room. What I want to accomplish would be successfully connecting my Smart TV and my Nintendo Switch, at the same time, via a cabled connection to my internet connection. My current issue is that, in order to successfully stream anything on my Smart TV without having to wait forever using our WiFi, I have my Smart TV connected to the open ethernet port on the back of the main Uverse DVR box that is in the den next to the TV. I do the same for my Nintendo Switch when I want to play games online. However, to utilize that same connection for the Nintendo Switch, I have to unplug the ethernet cable coming from the Smart TV from the Uverse DVR box and then plug the ethernet cable coming from the Nintendo Switch to the newly open port on the back of the DVR box that was originally being used for the Smart TV's streaming connection. This isn't a terribly awful setup, but it is frustrating at times because I would prefer not to have to switch out the cables everytime I want to switch devices. I have attempted using a "Network RJ45 Splitter Adapter, 1 to 2 Port Ethernet Socket Connector,RJ45 Male to 2 x Female LAN Ethernet Splitter Adapter Cable Compatible with Cat5, Cat5e, Cat6, Cat7 (Network Cable 1-2)" that I purchased from Amazon because, with my limited networking knowledge, I thought this would solve the problem. I plugged the splitter into the back of the Uverse DVR box and, then, plugged the ethernet cables, coming from the Nintendo Switch/Smart TV, to the two open ports from the splitter in an attempt to split the internet connection to both devices. Unfortunately, this attempt was a bust and the splitter only made it so that neither device would successfully connectto the internet and I was right back where I started.

My question is: Without utilizing my WiFi (because the connection is just not fast enough and never will be), what would I need to purchase to successfully connect both devices via a cabled connection to the one open port on the back of the Uverse DVR box?