jw5801 wrote:However this is resulting in only the second caption appearing under the first image, and the first caption not appearing at all, although both appear in the list of figures.
Then you have redefined the figure environment somewhere in your document. I assume you are using the float package and have the line \restylefloat{figure} in your document. If this is true, please read the documentation of the float package properly and you'll see that the above behaviour is totally correct in terms of the float package. If you don't know why you have \restylefloat{figure} in your document (because you started with a document scheme you got elsewhere), just remove it and this will solve your problem immediately. (And if there is a \restylefloat{table} as well, removed it, too.)
Attempting to use \captionof throws me an odd error about incorrect \prevdepth
It seems you have either a very early version of the caption package 3.0 or 3.1. (Please take a look at your log file.) Updating the caption package should help here. If not I'm very interested in getting a complete LaTeX document which shows the problem, so I can examine and solve the problem.
Axel