Yes, you can (and should) print the forms with no tax ID number on the return. If your children are receiving social security numbers, you will receive the full $2,000 child tax credit for qualifying children under the age of 17. However, if they will be receiving an ITIN, they will only be eligible for the $500 credit for other dependents. If the number you entered matches the sequence of an ITIN, that is likely why you see a change. (Your post masked the number, so I cannot tell if that is the case.)
You can file the tax return with your current address and then submit Form 8822, IRS change of address (click on this link for the form: Form 8822.) However, if you can access mail at the new address sooner than your move date, I would file using that address.
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