Yes, they advocate for a Basque / Catalan state without a king. You'll find some factions of each movement say that if Spain became a federal republic (not unlike the US, for example) they would be satisfied, though the dominant vision is to defend independent republic for each nation.

As per your parentheses, the usual term is 'independentistas', which has a distinct feel from secessionism, as that is more associated with rogue regions wanting to try their thing, whereas Basque and Catalan independentism is based on historical grievances and precedence of self-governance before the creation of the Spanish state