No, ciasteczko is a cookie (nominative and accusative singular). Ciazteczka is cookies (nominative and accusative plural).
The reason that ciasteczka also has cookie listed as a hint is because it's also the genitive singular form, meaning (of a) cookie. But the verb to have is followed by the accusative, not genitive case.
It's accusative. Mam (I have) + acc.; Nie mam (I don't have) + gen; Instrumental form is used after some prepositions, when one wants to say: I did it with a tool - name of that tool would be in instrumental or when one wants to say A is B, if B is a function, object (when B is a proper name we use nominative).