The Castle Hotel on High Street is one of the older inns in the walled town, and Dawson's, its restaurant, is a reliable choice for a proper sit-down meal in Conwy. The menu leans on Welsh produce — lamb from the surrounding hills, fish from the estuary and the Irish Sea coast, vegetables from local growers — and handles it without too much fuss.
It is the kind of place that works for a celebratory dinner or a post-castle lunch: formal enough to feel like an occasion but not so formal that you need to plan ahead. The bar is open to non-diners and serves a decent pint alongside the food menu. Booking is advisable in summer.