"Can you use Turbo Tax after the extension deadline?"
Yes, but e-filing closes after October 15, 2019 for the 2018 tax season.
If you cannot finish preparing your 2018 return by October 15, 2019 then yes, you will need to purchase and download the desktop software for 2018 to a full PC or Mac, and your return will have to be printed, signed and mailed. You will not be able to e-file.
If you owe tax due, then you will be subject to late fees and interest. TurboTax will not calculate that; the IRS will calculate any penalties/interest based on when they receive your return and payment, and they will bill you later on.
If you are getting a tax refund then there are no penalties for filing late.
If you do use the Online software to prepare...even if you print and mail it after the deadline....the Online software will shut down entirely sometime after the 25th of Oct (sounds like maybe the ~30th).
When that happens, your Online account will be locked entirely for a couple weeks or perhaps a month+, while they transition to the 2019 software.....during that lock-down period, neither the 2018 PDF nor the .tax2018 data file will be available for you to download, even if you do have the desktop software to continue working on it. You'll need the .tax2018 datafile to continue working on 2018 with desktop software......plan accordingly to get that datafile well before the lock-down if you have been using "Online" software for your 2018 data....or download a copy of the datafile after every session you enter some new data, starting say..the 24th of October.
Either that, or start working with the 2018 desktop/download software now....then it won't be an issue.