I’ve been to Seville and Cordoba many times over the past 15 or so years, staying about a month or more in Seville and Cordoba.
I stayed at Hotel Las Casas de El Arenal, in Seville, in February this year for 1 night, whilst our Airbnb was flooded due to a massive weather event. Possibly within your price range, booking.com. Excellent location. All locations in the old town are an easy taxi ride to Santa Justa. Taxis are well priced and reliable.
There is a mind boggling array of tapas bars and restaurants to choose from, though one we enjoy in Seville is Islamorada Tapas Bar in the El Arenal area, where we stay. For the BBQ style you mention we gravitate towards Moroccan restaurants, in both Seville and Cordoba, given the Moroccan influence in Andalusia, we generally find this type of food exceptional.
In Cordoba we enjoy El Rincon de Carmen, and La Posada del Caballo Andaluz, both are in gorgeous settings and the food is consistently good.