This problem appeared to boil down to a combination of two things:
- I made sure to get the latest version of Haxe and Haxelib.
- When I tried to trigger a build from Sublime Text 3, I had to make sure to have the project's build XML file open and active. Once it was the active file, I'd then choose a build target and start the build. After it had built successfully, I could change the active file and still complete a build, but sometimes it would appear to "forget" and I'd have to go back to the XML file, choose a build target, and trigger the build once again from the build XML file.