There was no clear answer to this problem (at least from what I read there). I'd like to post my experience with this problem so that in might help other people new to Processing.
It turned out that in my case this message "Could Not Find Main Class" when trying to run my exported application on another pc was do to the Windows file security feature, which "Blocks" files which had been downloaded from the web. This meant that I had a bunch of blocked files (Java files and Processing files which were downloaded from Processing and Oracle) on the pc that I had written and created my exported app on.
There are too many files to manually "unblock", so I found a free program that helps unblock all files at once, called "Alternative Stream Viewer". I went and unblocked all of the files in my Processing Folders and Java Folders.
I then went and re-created my exported app, zipped it and e-mailed it to my other pc. Unzipped, double clicked the .exe file and everything worked.
(I also ran the "Alternative Stream Viewer" tool on the Java folder in the pc that is testing the app (the pc that I had e-mailed my app to) and "unblocked some files there as well, before running the app.)
I'm a beginner with all of this, maybe someone with more experience can confirm this. I believe that because some files were being "blocked" on my development pc, these files were excluded in the exporting app folder?