Edited to add: You have the right website. I don't know whether the Alhambra itself has ever offered tours; the Nasrid Palaces are usually quite crowded, so it would hard to keep a group of any size together. There's also the issue that if you stopped to take photos, you'd probably lose touch with the rest of the group. I thought the audio guide was fine.
The Alhambra Complex is very large; that's one reason many people spend a good chunk of the day there. If you'd be satisfied with primarily seeing the Nasrid Palaces, you would need terribly long, I don't think. However, you need to be at the entrance point not long after your entry time, and the train station in Granada is not in the center of town, so it would take more than a few minutes to get to the Alhambra from the station. You'd need to figure what to do with your luggage, too.
It's really surprising to see tickets sold out this far in advance. I wonder whether the complex is going to be closed October 4-6, but I don't know why it would be.
The Alhambra General ticket is the one most people prefer. The Dobla de Oro General ticket (which includes some extra sights in Granada at modest extra cost) is a good back-up plan, as is the Night Visit to the Nasrid Palaces plus a separate ticket for a daytime visit to the rest of the complex (Gardens, Generalife and Alcazaba). The problem in this case is that all the alternatives are also sold out for October 4-6, even the Gardens, Generalife and Alcazaba tickets, which don't include the Nasrid Palaces and are very much an inferior option. I even checked a third-party website, and it had October 4-6 blocked off.
Something strange is going on, but I don't know what. If there's something special going on in Granada (a massive convention??) the middle of that week, rather than total closure of the complex, tickets may become available later if the middlemen who bought them up are unable to sell all of them.
I'm not optimistic in this case, but in the past the Granada Card has been a workaround when individual Alhambra tickets were sold out. The 48-hour and 72-hour Granada Cards cover daytime entrance to the Nasrid Palaces, the rest of the Alhambra and quite a number of other sights in Granada. Those cards are somewhat more expensive than the Dobla de Oro General ticket, and many visitors to Granada don't stay long enough to take full advantage of the city card, but it has historically sold out later than ordinary Alhambra tickets. Cards for October aren't on sale yet, and if there's something special going on at the Alhambra, it's quite possible they'll never put cards on sale that allow access between October 4 and October 6. Here's the website: https://entradas.granadatur.com/