How do I fix a leaky outdoor faucet?
Woodford Faucet is probably the place to go for the parts breakdown if I can read the handle right. They produce a kit specifically for repair. You'll have to look for better identification, but here's the Model 17 kit (round handle) which probably addresses all the issues. 



American MFG that's been around like forever and has always been good about repair kits. You'll recognize their key'd faucets if you've used campsites.

EPDM Packing that's under the Nut