How can I add a user as a new sudoer using the command line?

Just add the user to the sudo group:

sudo adduser <username> sudo

The change will take effect the next time the user logs in.

This works because /etc/sudoers is pre-configured to grant permissions to all members of this group (You should not have to make any changes to this):

# Allow members of group sudo to execute any command
%sudo   ALL=(ALL:ALL) ALL

As long as you have access to a user that is in the same groups as your "original" user, you can delete the old one.

Realistically, there are also other groups your new user should be a member of. If you set the Account type of a user to Administrator in Users Settings, it will be placed in at least all of these groups:

adm sudo lpadmin sambashare

Because your system configuration may vary, I suggest taking a look at the output of groups <username> to see what groups are normally in use.

I did

sudo usermod -a -G sudo <username>

as recommended here.