Question: In this Java program I'm writing I’m finding that I’m clogging up the `main()` method with a lot of code that would make it hard to quickly understand/read. Specifically due to a lot of error checking that I’m doing. As a result I want to write a quick function that is only going to be used by this one method to improve readability.

Is writing another method within the class my only option or are there other alternatives?

Answer: 
Answer is yes, generally `private` methods in the classes hold the code which is not used by outside world. But havig private methods help to reduce the cyclomatic complexity of your public methods. Smaller methods lead to more readable and understandable methods.
Since Java was designed that way, you can only have methods. So, yes, that's the only way. Usually you would use class methods (static) for instance independent methods. These can also be within another class, for example a utility or helper class.