This question was from the problem section of my textbook but I'm not sure how to start it.
At first, I thought it would just be like finding the sum of two variables which can be found in the textbook: https://i.stack.imgur.com/5HUZc.jpg
How do you deal with the random variables if they aren't independent? Should I just follow the example for the sum of two random variables but just stop right before the part where it assumes they are independent?
Yes, if they aren't independent, you can't go any further. Of course you will also have to modify the set $R_z$ from the textbook because they were dealing with $X+Y$ and you are looking at $X-Y$.