@jburatti
You are correct that NJ does not have a legal separation that is recognized by the IRS as eligible for using the HOH status. The requirement to live apart for the last 6 months of the year means starting on or before July 1. If you separated after July 1, 2023, then neither one of you is eligible to file as HOH.
Regarding claiming dependents, the rules for 2023 are different than the rule for the future. For 2023, both parents are eligible to claim all 3 children as dependents, since the children lived with both parents for more than half the year (from January 1 through the separation date in July). If you can agree on how the children should be claimed, then either of you can claim them. If you can't agree, the first tiebreaker is which parent did the children live with the greater number of nights. This is you per your answer. But for 2023 only, your verbal agreement is sufficient. (Note that when filing, each one of you will be asked about dependents and there are some questions about a custody agreement. Each of you should only list the dependents you agree to claim, and don't even mention the other children. Answer NO to any questions about custody agreements, because those questions only pertain to court orders between parents who live apart more than half the year.)
HOH is not allowed for either spouse.
If you can agree, you can file married filing jointly and claim all 3 children together. This usually results in lower taxes, but you may not want to entangle your finances. If you don't agree to file jointly, then each of you musts file MFS. That's what you need to do now. If your spouse filed HOH, that problem is between them and the IRS.
Starting in 2025, the rules on claiming dependents will focus almost exclusively on how long the children live in each home, with the parent where the children lived more than half the nights being the only parent with the automatic right to claim the dependents. See publication 501 under "special rules for children of divorced or separated parents."
https://www.irs.gov/forms-pubs/about-publication-501
