I solved my problem, though I can't pretend to understand how it came to be in the first place. For the reference of those who may find themselves in a similar position, I will post my solution here.
1. I copied the family in question and renamed the copied file with a 2 after the name. In this way, I could really get in and destructively experiment to discover the problem.
2. I deleted all the geometry and drew a box (the basis of the VAV box I was building) and added the length, width, and height parameters, which had maintained their values from the original family when I copied it.
3. Upon loading the family, I still got an error.
4. I deleted sections of parameters at a time, attempting to load the family into the project after each try, and found that by deleting all the parameters in the "dimensions" group, I prevented the problem.
5. I began deleting single variables in the dimensions group, and then realized that the problem could have been caused my any number of parameters. Sure enough, deleting the Unit Length and Unit Width parameters solved the problem.
Why?
Because when I deleted them, it came up with a warning box saying that the parameters being deleted were used in two dimensions (I had only used each one once). To prove this, I again deleted all the geometry, and then deleted those two parameters, which warned me that each was being used by a single dimension/array, even though there was no geometry at all. Whatever illusive value was using those parameters, it was acting at the broomstick in my spokes.
6. Since I couldn't find where those parameters were being used, I created two new parameters of the same name but in all lower case letters, and assigned values as such:
unit length = Unit Length
unit height = Unit Height
Next I hit apply, cleared the formulas, then hit apply again. I also looked for other formulas which used those parameters, such as:
Attenuator Height = Unit Height
and replaced the soon-to-be-deleted parameters with the new ones, like so:
Attenuator Height = unit height
Then replaced the dimensions in which I had used those parameters with the new, lower-cased versions. At this point, nothing that I needed was using the original Unit Length and Unit Height.
7. Knowing that deleting the original parameters would remove them from those phantom dimensions, I deleted Unit Height and Unit Length.
8. Next I renamed unit height as Unit Height and unit length as Unit Length, so that everything looked exactly as it had before.
And then it all worked. I used this method in the original family, and it worked there as well.
I do hope that this is helpful to someone.
