Migrating

karmenrose wrote:

The files do keep showing up on the new Mac. Are you saying I need to sign out of and not use iCloud on the new Mac?

Not exactly, but according to the linked article, "Turn on Desktop and Documents on every Mac that you want to use with iCloud Drive." IOW, turn it off on the "new Mac," but be sure to refer to this advice in the Support Article:

Turn off Desktop and Documents

1. From your Mac, choose Apple menu  > System Settings. In macOS Monterey or earlier, choose Apple menu  > System Preferences. 
2. Click Apple ID, then click iCloud.
3. Next to iCloud Drive, click Options.
4. Deselect Desktop & Documents Folders.
5. Click Done.

"What happens when you turn off Desktop and Documents

When you turn off Desktop & Documents Folders, your files stay in iCloud Drive and a new Desktop and Documents folder is created on your Mac in the home folder. You can move files from iCloud Drive to your Mac as you need them, or select all of your files and drag them to the place you want to keep them.

What happens when you turn off iCloud Drive or sign out of iCloud

If you turn off iCloud Drive or sign out of iCloud, a new Desktop and Documents folder is created in your home folder. You also have the option to keep a local copy of your files that are in iCloud Drive. If you keep a local copy of your files in iCloud Drive, they're copied to a folder called iCloud Drive (Archive) in your home folder. You can move any files that were in your iCloud Desktop and Documents to your new local Desktop and Documents."

karmenrose wrote:

How do I stop new files I make on my old mac from going to my new mac after I've stopped Migration Assistant?

After Migration Assistant has completed, the "old mac," and the "new mac" are independent of each other. Files created on either one will be isolated to that device.

Additionally, to ensure that files do not transfer, verify that you are not using iCloud Desktop and Documents.

What all is enabled in iCloud Prefs?