How do I change what item the verification code gets sent to when signing in to Apple?

Are you asking about the two-factor authentication code? If so, it should be sent to all of your "trusted devices" every time (although I have noticed that it isn't completely consistent in doing that and sometimes some trusted devices do not get the code). There is no option to change that or to select specific trusted devices. If you don't have a trusted device with you, you can have it sent to a trusted phone number on a case by case basis. There is more information in Two-factor authentication for Apple ID - Apple Support

re: "since I can't add my cell number for an alternate verification"

You can add any phone number including Androids and landlines and use them as trusted phone numbers. It is however a bit more difficult using them than trusted devices. There is more information in Two-factor authentication for Apple ID - Apple Support