Hi hobokelly123,
Thanks for posting in Apple Support Communities.
We understand that you're repeatedly seeing a sign in pop up for the Podcast app on your Mac. If you have not done so already, restart your computer to refresh running processes. Check that your Apple ID and password are correct by signing in to the Apple ID site.
Manage your Apple ID
If the Apple ID is correct and the issue continues after restarting, restart in safe mode and check for the same behavior and then restart to your normal desktop user.
Safe mode prevents certain software from loading as your Mac starts up. This includes login items, system extensions that aren't required by macOS, and fonts that weren't installed by macOS.
It also performs a basic check of your startup disk, similar to the more comprehensive check performed by the First Aid feature of Disk Utility.
And it clears some system caches, including font caches and the kernel cache. These are automatically created again as needed.
Before using safe mode
If possible, update macOS and your apps before using safe mode. This is more likely than any other step to resolve the kind of software issue that safe mode can help identify. If necessary, you can start up in safe mode and then update your software.
* Install the latest macOS updates.
* Install the latest updates for your third-party apps. If you got the app from the App Store, click the Updates tab in the App Store to check for updates. Or get the updates directly from the app developer or their website.
To be sure that none of your connected peripherals and accessories are at fault, test again after unplugging them from your Mac, including printers, drives, USB hubs, and other nonessential devices.
Start up in safe mode
Find out whether you're using a Mac with Apple silicon, then follow the appropriate steps.
If you're using a Mac with Apple silicon
1. Shut down your Mac.
2. Press and hold the power button on your Mac. As you continue to hold the power button, your Mac turns on and loads startup options. When you see Options, release the power button.
3. Select your startup disk. It's named Macintosh HD, unless you changed its name.
4. Press and hold the Shift key, then click the “Continue in Safe Mode” button below your startup disk.
5. Log in to your Mac. You might be asked to log in again.
If you're using an Intel-based Mac
If you're not using a Mac with Apple silicon, you're using an Intel-based Mac.
1. Turn on or restart your Mac, then immediately press and hold the Shift key as your Mac starts up.
2. Release the key when you see the login window, then log in to your Mac.
3. You might be asked to log in again. On either the first or second login window, you should see ”Safe Boot” in the upper-right corner of the window. If you can't start up in safe mode, make sure that your Mac isn't using a firmware password.
Use safe mode on your Mac - Apple Support
Regards.