Ticket options for sold out Alhambra dates

Ask your accommodation. There is always an allocation of small hundreds each day for the hostelries of the city.
Come on back if they are indifferent or unable.

There is a product sold at the tourist offices called a Bono Turistico which provides several values like bus rides and entry to the Alhambra. There are a limited number of alhambra slots available each day for this pass and you wouldn't buy it unless you could be sure they had 6 slots available that day they could book you into.

You could also contact your hotel as some hotels apparently have access to pools of tickets. That is the first thing I would do.

There are also walk up tickets available first thing in the morning on site but they do sell out, so I would plan to be there very early to line up and be at the front of the queue.

And there are guided tours which of course would be expensive for 6 but worth it. (I bought a season ticket to the Grand Palais art exhibit series last year as it was the only way to see the Hopper exhibit in Paris which was sold out the entire month of November I was there -- costly but worth it for me to see that particular exhibit. I expect that the tour price if the other options don't work, will be worth it for you too.)

We arrived in Granada without Alhambra tickets - because we often travel without reservations and weren't sure when we would get to Granada.

We followed the directions in Rick's guide. First to a bookstore that sells tickets - with no luck, not even for a night visit. Then to a tourist kiosk on the main street. They sold a pass and audio guide that covered many sights in Granada. They we're able to book us for 9 am the next morning.

Rick's book was worth the money, and the weight in my carry-on, just for his advice on getting tickets to Alhambra when you show up without advance reservations.