App Store keeps asking me to type in verification code after password, but I'm receiving no verification code
This happens because you have enabled 2-factor authentication on your Apple-ID (which is a good thing). Mavericks was created before this 2-factor authentication method was available for Apple-IDs and thus do not know about it. In order to still support logging in from Mavericks, you have to enter not only your password in the password field, but your password followed by the 2-factor verification code (6 digits).
The verification code is shown automatically on trusted devices (such as iPhones, iPads and Macs). If you haven't got a trusted device and are logging in through a web site, an iOS device or a recent macOS version, you'll be able to choose to have the verification sent as a text message or voice call. However, if you only have your Mavericks Mac, it is too old to support that feature.
I would suggest upgrading your macOS version to the latest possible version. 2-factor authentication is fully supported with Mac OS X El Capitan (10.11). Your MacBook running Mavericks should be fully capable of running El Capitan as well.