@taxsavvy , yes --- file the W-7 and the return together.
Since you owe to the IRS and State(?), it is a good idea to pay in by the deadline. Then the only issue would be if IRS disallows the ITIN and therefore your Child & Dependent Care Credit, you may a little bot more and that is a risk worth taking , esp. since by that time hopefully your mother has had the GreenCard. IRS is quite backlogged and the ITIN issuance may be as much as a year away.
Note that for paying your dues , TurboTax will only calculate the amount but has nothing to do with the payment thereof. You can either write a check and send to the IRS along with the payment stub produced by TurboTax OR go to the IRS site and there are options for payments -- credit card or direct debit from your bank etc. --- EFTPS. This would be better than writing a check. See here -->>
https://www.irs.gov/newsroom/how-to-pay-your-taxes
Namaste ji
pk
