Since you have mentioned 'Advaitavaad' in your question I'm answering from Adi Shankaras Advaita perspective.
Yes, Self realization doesn't imply Omniscience.
It is because in Shankaras Advaita the attributes like Omniscient and Omnipotent etc... seen in Brahman are due to the product of Avidya. I discuss it in my answer here. Shankara in Brahma Sutra Bhasya 2.1.14 states:
तदेवमविद्यात्मकोपाधिपरिच्छेदापेक्षमेवेश्वरस्येश्वरत्वं सर्वज्ञत्वं सर्वशक्तित्वं च, न परमार्थो विद्यया अपास्तसर्वपाधिस्वरुपे आत्मनि ईशत्रीशितव्यसर्वज्ञत्वादिव्यवहार उपपद्यते, तथा चोक्तम् - 'यत्र नान्यपश्यति नान्यच्छृणोति नान्यद्विजानाति स भूमा इति' यत्र 'त्वस्य सर्वमात्मैवाभूत्तत्केन कं पश्येत्' इत्यादिना च एव परमार्थवस्थायां सर्वव्यवहाराभावं वदन्ति वेदान्ता ।। 2.1.14
Hence the Lord's being a Lord, his omniscience, his omnipotence, &c. all depend on the limitation due to the adjuncts whose Self is Avidya; while in reality none of these qualities belong to the Self whose true nature is cleared, by right knowledge, from all adjuncts whatever. Thus Scripture also says, 'Where one sees nothing else, hears nothing else, understands nothing else, that is the Infinite' (Ch. Up. VII, 24, 1); 'But when the Self only has become all this, how should he see another?' (Bri. Up. II, 4, 13.) In this manner the Vedânta-texts declare that for him who has reached the state of truth and reality the whole apparent world does not exist.
Thus in Advaita these qualities like omniscience, omnipotence are the products of Avidya. So they do not exist in highest state. So, there is no meaning of omniscience and omnipotence to a realised soul. However one can get powers of omniscience through Tapas, Yogas, Yajna phalas etc...
As a sidenote the BrihadAranyaka upanishad verse which Shankara is quoting there is 2.4.14 which states:
II-iv-14: Because when there is duality, as it were, then one smells something, one sees something, one hears something, one speaks something, one thinks something, one knows something. (But) when to the knower of Brahman everything has become the self, then what
should one smell and through what, what should one see and through what, what should one hear and through what, what should one speak and through what, what should one think and through what, what should one know and through what ? Through what should one know.