Pretty easy. 1. Pistol in hard case that you only have the key or combo to, NO TSA locks. 2. Ammo in original packaging or MTM Caseguard type boxes, NOT IN magazines. 3. You will have to look up the max amount of ammo you can take. 4. Go to the counter/kiosk, tell the nice folks you have a firearm to check. 5. Fill out the nice form the folks give you. 6. Go to the other end of the airport for the special firearms screening booth. 7. Wait until they are done with it and go to your plane. I just did this for a buddy who was taking an AR and an 870 to Pheonix a couple weeks ago. As painless as can be expected.
If you know someone in Arizona you could ship it to yourself care of them. I don't know if you could ship it straight to them or if you'd need to use an intermediary such as an FFL but it's worth looking into as well I think. Perhaps an FFL on the board here could clear that up.
Bolded part of your quote is dependant on airline. Southwest doesn't care whether it's in the locked case or not. But magazines/clips must be unloaded. Best advice I can give is follow the TSA regs. Have a printed copy of them and what ever airline you are flying's policy with you when you get to the airport.
I just flew back from Florida last week on Continental. I put my P220 in a blue SIG plastic case after removing the ammo and mags. I put a lock through each of the two locking holes in the case. I put each of the empty mags in separate zip-locs. I placed the gun case, holster and magazines in my checked baggage. When I got to the Continental counter, I told the girl that I wanted to check one bag and declare an unloaded pistol. She asked to see the case and I had to fill out a tag that we attached to the case. I had another lock run up the magwell and through the action (just in case) but she only asked if it was unloaded and then never asked my to open it. I then placed the guncase back in the bag, took it to the TSA baggage check and informed the agent of the situation. He took it right to the machine instead of throwing in a pile or on a cart. After it went through, he came over and said that everything was OK and I left for the security check and the gate. Being my first time, I was a little anxious about this but everything went very smoothly. I did not take any ammo with me, so I cannot speak to that. You might also consider placing the gun in some sort of dry plastic bag, a produce bag should work here, so that everything in your suitcase doesn't smell like Hoppes No 9 when you get there. Good luck and check your airlines website ahead of time.