All methods that update the GUI must be invoked on the EDT, otherwise you may end up with some unexplained GUI behavior (which sounds like what you are seeing). You could have repaints that don't happen properly, thread races, etc.
It is not advised to run long running tasks on the GUI because they will cause the GUI to become unresponsive, so for long running tasks, SwingWorker is a good solution to use (note that the process and done methods are called on the EDT automatically so your worker can do its long running work in doInBackground but you can safely update the GUI without using SwingUtilities.invokeLater from the done method).
As you mentioned you have hundreds of methods and you don't want to call SwingUtilities.invokeLater every time, you might want to look into one of the task frameworks. The Swing Application Framework was developer under JSR-296 http://java.sun.com/developer/technicalArticles/javase/swingappfr/ but is not actively supported, but still provides a nice framework.
http://en.wikipedia.org/wiki/Swing_Application_Framework is a list of alternative frameworks.
It sounds like you are going to need to do some significant rewriting in your application. Calling GUI methods from outside the EDT is unsafe to do.