Never mind. I figured it out.

In order to be able to access my class from multiple instances of my application, I need to not only create it with the OLEPUBLIC clause. I also must go into the Project Information window under the Servers tab. All of the OLEPUBLIC classes show up there as server classes. I have to designate the class that I want to be shared as "Multi-use". Once I've done that, I can use the GETOBJECT() function to look for an existing instance of the class. Each instance of my application shares the same instance of the class.