There are many and various types of beef, pork, and fish. Stews are very common. Fish is very often pan fried without flour - olive oil, butter, garlic, white wine is normal. I managed fine without much compromise. The Cadiz area is big on potatoes as a side dish, so I found it easy to just leave those be. My companions benefited.
This restaurant
https://www.cumbresmayores.com/ is traditional and very busy in December last year. Ate there twice.
The market is superb and has lots of cafes and smallish restaurants nearby
https://www.mercadosdecadiz.com/
The old school sherry bar in Calle Feduchy is called Taberna la Manzanilla and is recommended if your carb budget will stretch to a Pedro Ximinez or three, it's worth it.
There was an excellent mainly fish restaurant too (there are many) but I can't remember what it was called....
Chicharrones (pork scratchings) usually avavilable as snacks.
Hope that's a start.
Edit:
https://maps.app.goo.gl/4k7mbhA4pQz3W6eJ8
This is the fish restaurant. It gets really busy but was worth it. There are also a few good beer bars (for those that drink beer) - one being Cerveceria la Barra on the Plaza de San Juan de Dios.