Hi Eddie!
Replacing the joystick should fix it! But most of the time, it's soldered on... So you need to de-solder it out and solder the new stick in.
If you have experience with soldering, then it shouldn't be too difficult.
And while you are there, I recommend to replace it for the hall effect joysticks! They say that's impossible to have drift with those!
Here is a guide: https://www.youtube.com/watch?v=VhjanOAMCvo
Please let us know how it goes!