No. You do not need to file a form 944. Since you are the parent of the children under age 18 and your business for tax purposes is a sole proprietorship, the children are not subject to social security or medicare taxes. Further, since they are under age 21 they are not subject to unemployment tax. Additionally, since the pay is under $12K there would be no income tax withholding.
Here is the IRS reference.
Having said that you should issue forms W2 with a copy to them showing the salaries paid. This supports your business deduction for the salaries paid. Again the W2s will not have any social security, medicare or income taxes withheld.
You could issue a 1099-NEC instead of W2 however then the children would be subject to self-employment taxes (social security/medicare).
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