Apparently, to submit a complaint to the CEDAW one does need to exhaust national remedies.
https://www.ohchr.org/en/special-procedures/wg-women-and-girls/note-different-procedures-within-un-system-dealing-womens-human-rights-violations
There exist several avenues for bringing individual complaints relating to discrimination against women to the United Nations bodies. One can send a complaint to the Committee on the Elimination of Discrimination against Women or another treaty body, such as the Human Rights Committee, the special procedures of the Human Rights Council or the Commission on the Status of Women..
Generally, complaints should include a detailed factual account of the violation and all supporting documentation. They should not be politically motivated nor have abusive language. Each complaint system has its own advantages, but some mechanisms will not accept a complaint if another is already dealing with the issue, so it is important to understand how each mechanism works.
Committee on the Elimination of Discrimination against Women
The Committee on the Elimination of Discrimination against Women (CEDAW) is the body of independent experts that monitors implementation of the Convention on the Elimination of All Forms of Discrimination against Women. CEDAW Committee consists of 23 experts on women’s rights from around the world.
Victims of an alleged violation of the Convention on the Elimination of All Forms of Discrimination against Women (CEDAW) may submit a complaint to the Committee on the Elimination of Discrimination against Women. The procedure under the Optional Protocol is quasi-judicial and if successful, the Committee will request the State party concerned to provide redress to the victim.
The complaint will only be admissible if:
The State party responsible for the violation has ratified or acceded to an instrument called the Optional Protocol to CEDAW (OP-CEDAW) before the violation took place or, in the case the violation took place before such ratification or accession, it can be successfully argued that the violation has continuing effects, i.e. it continues to deploy its effects after the entry into force of OP-CEDAW
The victim has “exhausted domestic remedies,” meaning that the victim has tried to remedy the violation in her country first. This requirement can be waived if the domestic remedy will be ineffective or take too long to realize.