The picture posted by Wynn was an exact match to the infection.
As much as I hate to even though it had not flowered for me, yet, I've decided to get rid of the plant. If they were mealybugs, I likely would have tried to save it. However, from what I read, this particular type of scale is nasty and very difficult to eradicate.
I bought the plant from Lowe's. It was a gro-more plant, very inexpensive at under $13, and my guess is that it was already infected when I got it since the infection is so severe. I think it would take more time and money than I paid for the plant to nurse it back to health - as much as I hate to put it that way.
I hate getting rid of plants for any reason.
I also think that I would be endangering our other houseplants, too, by keeping it; therefore, it seems too risky to keep.
We had an outbreak of aphids about two years ago which spread to many of our house plants. We eventually were able to cure the outbreak by closely inspecting the plants that were the most susceptible to the infection, and picking off the aphids with tweezers, alone. I still had to get rid of one plant, however, we were lucky enough to be able to save a few non-infected clippings. That was a painful enough experience, and I do not want to go through that again with a bug that sounds as difficult as scale is to eradicate.
On closer inspection, the dried coverings that others said to remove are all filled with the bugs.
Anyway, now I know what to look for when I buy more orchids in the future.
That said, I bought two other orchids at the time (also gro-mores) that are doing fantastic. Another cattleya and a dendrobium aggrgatum. They each have no signs of infection, and I hope to keep it that way.
Although this was a difficult decision to make, thank you to everyone for the help.