DxDiag is not reporting the presence of the Radeon card.
What do you physically see in the Device Manager ?
Device Manager only shows Intel HD Graphics 4000.
I actually have an update since posting this thread. I realized that my motherboard has two PCIe ports, so I switched my card to the other one and this fixed the issue briefly. My monitor recognized my GPU and I was able to install the drivers without any errors. It rebooted normally but after a few minutes the screen went black, and now neither port will recognize the GPU. Also, sometimes the fans will not spin when plugging it in, and sometimes they spin too fast as if I'm running a game. It appears my last step might be to try the card in a different system, though the only other desktop in the house is a prebuilt Gateway.