The Fraud Alert will have no impact on anything outside of applying for new credit accounts. It will not prevent you from opening any new credit accounts either. It simply requires the credit issuer to contact you and verify your identity prior to processing any new accounts. This means if you do apply for any credit online you will not receive an instant approval but will see the "We need additional time to process" message and will need to contact them or wait for them to contact you.
I would maintain whatever credit monitoring service you are using and keep an eye on your reports. Hopefully, it was just a case of Consumer Cellular entering the wrong information instead of true identity theft. Even if it was ID theft the fraud alert and the failure of the initial try at opening an account should discourage them. If you are truly concerned you could request a credit freeze be placed on all your reports. This means that you have to contact the individual bureaus and provide them with a PIN and authorization to allow a specific creditor to receive information.
Starting Score: 500's