If short on time, and had to choose, I'd say, "New Orleans."  I've been all through Texas, New Mex, and Tennessee, and New Orleans is definitely going to be your best best for a short trip...   Texas is HUGE, if you want to see Texas you need a month..  If you want to see Dallas you need 10 days...   New Orleans you can see in about 4 days...   Memphis you can see in two days..
 
Btw, I am still trying to figure out the difference between Arizona and New Mexico..  I think it bakes down to Billy the Kid vs Tombstone...

And if ya come back through Arkansas you could stop over in the capital city and I'll treat you two to a killer diller dinner on the back deck.  Nawlins is a wonderful place.  Bring your appetites cause eatin is what we do best in the south! 

bayoubill
Why Am I so turned on by Miss Pussycat's photo?

 
I'm guessing it's the puppets. :-)

That picture brings back memories.  I still remember seeing a Kukla, Fran and Ollie show back in the early 60's.  The musical guest was Herb Albert (with his band).  This was down in Old Town San Diego (California).
 

I don't know much about LA. I do know that you can take a riverboat trip up the Mississippi river to Memphis. I think that would be cool... then reat a car and drive back to Vegas.
 
J