Earl,
The phenomenon you've stumbled across is a result of the fact that -- as good as camera imaging sensors have become, they still can't hold a candle to the human eye and its ability to "see" a WIDE dynamic range of color intensities throughout nature.
In the camera world, if you expose for the highlights in your image, then the a large degree of the layout goes dark -- being underexposed. And if you expose for the shadows in your image, then the light bulbs wash out -- being overexposed.
You really have two choices: either you REDUCE the dynamic range of light intensities in your image -- i.e., as someone suggested dimming the border lights. OR you can employ a technique used by professional photographers known as HDR (high dynamic range).
When using HDR, you "bracket" the exposures of your image using a range of shutter speeds with a given aperture. Best to have the camera on a tripod for this. Then later -- using software -- you combine the various exposures of the exact same image (that's where the tripod comes in very handy), so that highlights aren't blown out and shadows retain their detail. This technique is often used by pro photographers when capturing architectural images around dusk -- when they want to retain detail of a buildings exterior along with a colorful twilight sky, yet they don't want to blow out the highlights of windows being lit (often with warm tungsten lights) from the interior of the building.
Search "HDR photography" on the internet, and you'll find a bunch of hits worth reading.
Best of luck... and have fun expanding your photography horizons!!!
David