Okay fellow Apple-Addicts, I finally found the solution! - on my own. If you're determined to find a solution to a problem, click on stuff - do this for a few hours. Can't find the solution? No worries - just go ahead and click on the same stuff again for a couple more hours. Eventually, the answer finds you.
Here's the solution, in case anyone else goes through this same ****:
1) You get your new iPhone and it first goes through all the usual set up (What country? What language?, etc)
2) You come to a window that asks "Do you want to transfer all your apps and stuff from your old phone to new?" You figure this is a no-brainer, the only answer to this - of course - is "YES!"... right?
****LoUd-BuZzeR!!**** NOPE! WRONG answer!!!
You are now required to retrieve the verification number off your old busted, non-working phone. Ooops, you say...that won't work, so you try to go back and change your answer to "no" but now Apple thinks your screwing with them and their security kicks in. Now you no longer have the option to change your answer. Grrrr...
3) SOLUTION!: Go back to the first, "start setup" window and start going through the setup steps, only this time, somewhere along the way, you'll find the choice: "Go Back to Factory Settings". I don't recall exactly where this is because I clicked on so many things, but you'll find it if you just click on stuff. Since it's only a couple pages into the setup process, it won't be hard to find. So I clicked "Go Back to Factory Settings" and it wiped the phone clean and gave me a fresh start.
4) Now, when I get to the window that asks "Do you want to transfer all your apps and stuff from your old phone to new?" I was able to answer "NO!...**** NO!" So it then opened up the home screen where I was able to access phone, texts, etc.
5) Now that I had access to the home screen, I was able to have my carrier service switch over my service plan from old to new phone. Now, when Apple texted me my verification code, it came to my new phone. Done!
There is nothing on Apple Support FAQ's that tells you how to do this. You'd really think this would be a common problem among users. You drop your phone in the lake, it gets run over by a car, or the battery no longer charges so it's dead. So now, how do you get all your stuff onto a new phone without the old phone working? If you're old phone is not working, DON'T say "Yes - transfer my stuff please!" Just SAY NO! ...or else it will bind up the process. Don't bother calling Apple either, just do the super-secretive hack described above.
Again, thanks to all who responded.