Mike had everything right. Modern auto-everything cameras are the enemy of good TV photos. I use my digital camera constantly, but, to be honest, I took better photos 18 years ago when I launched my website and used a 35mm film camera.
Most people pay far too little attention to lighting. In addition to turning off the flash, I turn off the overhead room lights and use several portable lights to flood the area with
ambient (i.e., reflected and non-glaring) light. Backlighting is especially important. If you look at the typical rotten TV photo, you'll see far too much light on the front of the TV and none, or almost none, behind it.
As you've already learned, when presented with a bright screen in a comparatively dim background, your auto-everything camera will pick one of the two extremes to set the exposure. If it picks the screen, the screen looks OK and the cabinet (and the rest of the room) is too dark. If it picks the room, the screen is too bright.
It may help to decrease the contrast between the screen and the room. That means turning down the screen brightness and "turning up" the ambient lighting everywhere else. Even then, it may be a challenge to force your auto-everything camera to reproduce what our eyes see naturally.
When I am feeling impatient (which is most of the time), I simply combine two shots taken with the camera on a tripod. In the first one, the screen is turned off and the room is lit so that the cabinet looks best. In the second one the lighting is optimized to make the screen look best. Then you copy and paste the best screen into the best-cabinet photo. You could consider this a trick photo, but I'm really just compensating for what my stupid "smart" camera does against my wishes.
Color is a whole 'nother issue. Indoor lights are typically much yellower than daylight, so when my auto-everything camera adjusts that to make the room look natural, the TV screen becomes too blue. You can minimize this problem by getting bluish light bulbs from a camera shop and using them in floodlights. Or, if you are combining two shots, you can adjust the color balance of the screen portion to match that of the room.
Sometimes it's just too much hassle to try to get a single photo that shows both the screen and cabinet at their best. There's nothing wrong with using two photos. To display your beautiful cabinet, leave the TV turned off. To show your TV's beautiful picture, turn down the room lights and zoom in for a closeup of the screen.
Phil Nelson
Phil's Old Radios
http://antiqueradio.org/index.html