Can I still file my federal and state (NY) today (October 15th) electronically through my TurboTax Deluxe software or do I have to file paper copies and postmark them?

You can e-file up to midnight in your time zone on Tuesday, October 15, 2019.

Your extension gave you extra time to prepare your 2018 tax return documents.  It did not give you extra time to pay.  If you did not pay your estimated 2018 tax due by April 15, 2019 then when the IRS receives your 2018 return and your payment, they will calculate your penalty/interest and bill you later.  TurboTax will not calculate the penalty or interest.

You have until October 15, 2019 to e-file your 2018 tax return.  After that, e-filing for 2018 will be shutting down, and a 2018 return will have to be printed, signed and mailed instead.  The online software will also be shutting down, so if you are not finished preparing your 2018 return by then, you will have to download desktop software to a full PC or Mac to finish the return.  If you e-file on October 15, and your e-file is rejected you will have until October 20 to fix it and re-file it.

When you requested your extension you were supposed to pay your estimated tax due.  Did you enter the payment you made into your tax return?  Go to Federal>Deductions and Credits>Estimates and Other Taxes Paid>Other Income Taxes  This will show up on line 17 of your Form 1040

Of course you can file both your federal and state returns electronically BEFORE midnight today. In fact, filing electronically is the fastest and safest way to both file and pay any balance due (by bank account or credit card). You can also file by mail, but you'd have to print out all the documents for both filings, include checks to pay any balance due AND take them to the post office (properly addressed to the correct filing location) to make sure they are postmarked with today's date. If you opt to mail your returns, be sure to include the filing form that will also print out. It has the address you should use for that return.