IRS has an "installment agreement" where you can make monthly payments. They won't let you wait until next year to pay them. When you get to the File section of TurboTax, you'll be asked how you want to pay your federal taxes, and there will be an option for I want to request an installment payment plan. Follow the prompts and TurboTax will walk you thru the setup. IRS charges a fee to set it up.
IRS also has a 120-day plan, without a fee, if you can pay your entire balance within 4 months. Here's a link to the IRS payment plan page: https://www.irs.gov/taxtopics/tc202.html.
As for your state amount owed, most states have something similar to the IRS monthly plan, but you'll need to contact them directly. You can look up your state's Department of Revenue online. When you're in the File section of TurboTax, you'll also be asked how you want to pay your state taxes, and you can select I will mail a check. That will result in the return being filed, but you won't pay anything. You'll then have time to contact your state.
Similarly, if you want to use the IRS 120-day plan instead of the installment agreement, you'll set that up yourself online. In that case, choose I will mail a check when asked about paying the federal taxes.
