Hey LukeGaucher,
We understand that you're having trouble signing into Apple Music on a new computer, and we'd love to help out.
Just to confirm, does this only happen when you try to access Apple Music on your new computer? In other words, are you able to access Apple Music successfully on other devices without an issue?
Are you using these steps to try and sign into your Apple ID account on your new device? Sign in with your Apple ID
When trying to sign in, are you prompted to enter a verification code? "Whenever you sign in with your Apple ID on a new device or browser, you'll confirm your identity with your password plus a six-digit verification code. There are a few ways you can get a verification code. You can use the code displayed on your trusted device, get a text or phone call, or generate a code from your trusted device."
More about verification codes and signing in with two-factor authentication here: Get a verification code and sign in with two-factor authentication
Thanks!