When trying to reset my login / password, text in red says not connected to internet please verify connection, when the macbook air IS connected to wifi.

Question

cannot reset forgotten login password.

Answer 1

You don't need to be connected to the Internet to reset the password.
Note: If you've activated FileVault, this procedure doesn't apply. Follow instead these instructions.
Start up in Recovery mode. When the OS X Utilities window appears, select
          Utilities ▹ Terminal
from the menu bar at the top of the screen—not from any of the items in the OS X Utilities window.
In the window that opens, type this:
resetp
Press the tab key. The partial command you typed will automatically be completed to this:
resetpassword
Press return. A Reset Password window opens. Close the Terminal window to get it out of the way.
Select the startup volume ("Macintosh HD," unless you gave it a different name) if not already selected. You won't be able to do this if FileVault is active.
Select your username from the menu labeled Select the user account if not already selected.
Follow the prompts to reset the password. It's safest to choose a password that includes only the characters a-z, A-Z, and 0-9.
Select
           ▹ Restart
from the menu bar.
You should now be able to log in with the new password, but the Keychain will be reset (empty.) If you've forgotten the Keychain password (which is ordinarily the same as the login password), there's no way to recover it.