Quick add-on regarding flights and trains...
We are scheduled to land in Madrid at 9:20 AM on Saturday, with our flight to Jerez de la Frontera landing at 1:05. Would it be better to just skip that final flight and take the high-speed rail straight to Seville?
Also, if we opt for the Madrid route, is there any issue skipping our first flight that is scheduled to leave from Almeria and just hop on that second flight leaving from Madrid?
Finally, are rail tickets something we need to purchase in advance, or can we wait to make sure our flights are arriving on time and then purchase soon before needing to board? Thanks!

I clocked that in your question and I don’t think you can do that. If you can’t confirm it with your airline, I would not attempt it because your other legs could indeed be cancelled.
If you cannot pay to change the flights to what you want, I would make the best of where you do land, there’s no shortage of great places.
I don’t think I’d worry about booking any trains in January.

I’m surprised you just didn’t book a flight to Madrid then take the train to Sevilla. Flying to Jerez de la Frontera leaves you with a Madrid layover and a one hour commute from Jerez to Sevilla.

A two hours drive from Granada to Almería makes no sense if the only reason you’re going to Almería is to fly out. If flights from Granada didn’t work, Málaga is a little closer than Almería and has more to see.

With only a week in country, maybe just visiting Sevilla and Madrid with one possible day trip to either Córdoba or Toledo would simplify your plan. If you already have your flights, then you’ve really complicated the trip.

Your trip is quick and arrival/departure locations non-typical. Hands down, for us, the wow site in Andalucia is the Mezquita in Cordoba (plus the city is lovely and food excellent). That said, perhaps you could do something like this:
Saturday - arrive to Seville
Sunday - Seville
Monday - Seville
Tuesday - to Cordoba (Mezquita)
Wednesday - to Granada
Thursday - Granada (Alhambra)
Friday - to Almeria
Saturday - depart

I'd go to Madrid. Tons of amazing restaurants and the overall food scene is world-class. The Prado, Royal Palace, Reina Sofia, the Retiro, it's great. But, it feels more like a European capital than the fairy tale medieval Spanish citadel or Andalusian white washed hill town, so you have to be on board for that.

Thank you for all the advice. When I initially booked the flights, I thought we were flying into Seville and out of Granada, and only realized afterwards that AA auto-selected "nearby" airports. I booked in haste as we have struggled to make it away for a vacation since having kids and I wanted to feel committed to something.
Fortunately, the tickets are fully refundable if I can't make changes, and it appears I could just fly round-trip to Madrid for a few more miles.
As it's our first Europe trip, I think simply being in a new continent will be awe-inspiring alone, so although the Alhambra may have been what I was most excited to see, simplifying logistics and maximizing our time is most important.
With that in mind, I'm now thinking of flying into Madrid early Saturday, staying until Monday, at which time we will rail to Seville, then returning to Madrid Friday before our departing flight Saturday.

Use the extra points and fly in and out of Madrid. That will greatly simplify things.

With that in mind, I'm now thinking of flying into Madrid early
Saturday, staying until Monday, at which time we will rail to Seville,
then returning to Madrid Friday before our departing flight Saturday.

Instead, consider hopping on a train to a Sevilla when you land. That way, you will have two stays instead of three. I just find this more efficient. Time tends to slow down when you get away, have a great trip!

My main reason for considering staying in Madrid rather than taking a train straight to Seville is the concern of a flight delay causing us to miss a train booked in advance, as well as needing to get from the airport to the Atocha station with luggage after a long trip, before hopping the train to Seville.
If I could book upon arrival or have flexibility in train tickets should there be a delay, this option would definitely be preferable to me. I just have no Europe or train experience, so I'm unsure how difficult this would be.

Certainly solicit feedback from others, but I don’t think a train is likely to be sold out, maybe share your specific dates? And yes you do pay more. Overall, the discomfort in day 1 would be worth it to me, I just nap on the train, but weigh your journey and your wherewithal to do it.