Yes, the Democrats should give up on an entire region of the country....
That kind of arrogance worked very well for the GOP, oh yes.
Well, it kind of did. There's a reason the GOP did just fine without paying attention to the South for decades. If the Republicans want to appeal overwhelmingly to Southerners, the Democrats should try to appeal to everyone else rather than fighting the Republicans on their strongest turf for no particular reason. (Similar, of course, could be said in reverse during the late 19th century into the early 20th century.)
It is not wise for the Democrats to compete in the South when they can compete more effectively elsewhere. The Republicans dominate or are competitive a number of other states in which it is far easier for the Democrats to make inroads. Moreover, "Southern culture" is generally speaking not appealing in such states. So, by ignoring the South while the Republicans cater towards it, the Democrats improve their chances in these areas.
Realistically, the Democrats should only make an effort in the South where things appear to be moving in their favor--North Carolina and Virginia, obviously, and in the long run Georgia and possibly South Carolina. Incidentally, these are also the Southern states which are growing. The growth in the South is not so beneficial to the South as a political bloc as it might at first appear. The growth is fragmenting the traditional voting patterns, and it has already allowed Virginia and North Carolina to vote for a black man for President. The growing states are certainly therefore somewhat more worthy of the Democrats' notice--but the key to the Democrats gaining ground in the growing states is increasingly, not appealing to traditional folksy Southern politics, but offering the same appeal they offer in the Northeast, Mountain West or Pacific, since the new voters are generally speaking more "Northern" in political culture than Southern.
Texas is an interesting case, and in the very long run is probably also worth Democratic attention. But demographic patterns that far in the future are difficult to predict, to say the least. If the United States' economy never returns to its previous dominance, Mexican immigration will probably decline substantially.