This is what it means literally, as it stands in the Latin. There is no intrinsic semantic difference between 'se' and 'sese', both being standard classical forms of the 3rd person reflexive pronoun. The choice depends, as it depends in Cicero and in other authors who pay attention to euphony, upon the stylistic effect (the rhythm) at which one is aiming.

On the other hand, in the best Latin 'se(se) dedere' is primarily a military term and is thus closer in meaning to English 'surrender' than to 'give up', which latter has a host of connotations, often personal, which are generally wanting in the Latin phrase. 'Deponere' ('to give up', e.g., give up hope, give up on some attempt) is probably a more appropriate rendering.