Have you tried resetting your computer's System Management Controller (SMC)?
For the MacBook Air, here’s what you do:
- Shut down your computer
- Keep the MagSafe adapter (power cable) plugged in.
- Press at the same time shiftoptioncontrol (on the left side of the keyboard) and the power button
- Let go
- Turn your computer back on with the power button.
Also, another thing worth doing is resetting your NVRAM.
To reset your NVRAM do this:
- Shut down your machine. Yes, a full shut down, not just logging out.
- Press the power button and then press the commandoptionpr keys. You have to make sure you press these keys before the gray screen appears or it won’t work.
- Hold those keys down until your Mac reboots again and you here the startup chime.
- Let go of the keys and let your Mac reboot normally.
Note: When you log back in you may need to readjust some of your system preferences (e.g. mouse speed, time and date/timezone, etc).
Since we've established it works in Safe Mode, this means we've eliminated both the possibility of a network/router issue or a hardware issue. This is good, because it must be related to some sort of 'software' or 'data' issue. There is a remote possibility it will now work when started up normally, as booting in Safe Mode will have cleared some data.
If it doesn't, as another test, try creating another User Account and testing that. To do this, follow these steps:
- Go to System Preferences
- Click on Users & Groups
- Click on the + sign at bottom left (you may need to unlock the padlock before you can make changes)
- In your Login Options turn off Automatic Login and also ensure you display your login window as a list of users
- Exit System Preferences
- Shutdown your computer
- Reboot your computer and then login by selecting the User you just created
- Try connecting to WiFi
Please report back on the outcome.