We were in Seville for a few weeks last year, and we stayed in an Airbnb right next to the market in Triana. Triana is right across the river from the major tourist sites. From Triana you would take a very short walk across the historic bridge right into the tourist area. We loved our time in Triana. It's more local residential and less touristy than across the river, although a lot of tourists stop on the bridge and take selfies of them and the river. The river walk in Triana and across on the other side is very nice. And there are some great restaurants in Triana, although we mostly bought a lot of fresh fish and fruits and vegetables in the Mercado de Triana and the local grocery stores to eat. So for Seville, I highly recommend that you stay in Triana.
I don't know about Valencia, I haven't been there yet, but we are going later this year.