For recovery, all you need to know is your password. In a new computer, download AxCrypt (if you’d like to feel extra-extra secure, keep a copy of the installer around on a USB stick), install it and enter your email and password. You’re back to where you were.
Even if the AxCrypt servers are entirely unavailable for whatever reason, all you need is a copy of the software and to know the password.
The AxCrypt ID key pairs are for convenience when changing passwords and for sharing files with others. They are not required to open your own files. For that, all you need to know is the original password used when the file was first encrypted.
When you share encrypted files with others, you’re not sharing using your own key ( crown jewel ;-) ). You’re encrypting the unique encryption key for that particular file with the recipients public key part of their AxCrypt ID. You’re not sharing using your own AxCrypt ID. That’s for others to share with you – and vice verse.