The parent who can claim the children as dependents is the parent with whom the children spent the larger number of nights. You actually have to count the nights the children lived in the home of each parent.
It sounds like both parents get credit for 340 days (from 1/1/21 through 12/7/21). Then, for the rest of 2021, count where the children spent each night. Whichever parent had actual physical custody of the children for the night (in the home, or an apartment, or hotel room, it doesn't matter) gets credit for that night.
The parent with the most nights can claim the children as dependents. In the event the number of nights is exactly equal, the parent with the higher income can claim them.
You do need to file MFS, you can't file single or head of household.
If your spouse also claims the children and e-files first, you will be blocked from e-filing. If you have custody more nights, you can file your return by mail and get your refund. The IRS will send letters to both parents asking for more information but this usually takes 9 months to a year to get going.
