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In principle the taxis in Sevilla are not too expensive. However, there is a flat rate for transfers from the airport to the city. The rates varies from 21.89 EUR to 30.50 EUR according to the period of travel. You are strongly penalised by the application of this flat rate: it more or less doubles the fare in your situation!
The only public transport option to get from the airport into Sevilla is the line EA.
As an intermediate solution you might consider to take the bus EA to the Santa Justa train station and proceed from there by taxi, for 10 EUR or less.
You should be able to answer your last question by yourself.
Or should I just skip the nice hotel out there, and stay in the city centre within easy walk of the airport bus?
I would stay somewhere in the center. That way you minimze the travel and transfer times and you will get the most out of your weekend. You should not only factor in the price difference for the airport transfers, but also the daily public transport between the hotel and the sights, as well as the time loss.
Remark 1: there is a nice hotel (of a Spanish chain) at Plaza de Armas. It's next to the terminal of the EA bus line and only a stone's throw from the city center.
Remark 2: If you want to spend a weekend in a nice hotel with nice amenities, there are other better suited destinations for that!