Can I file my taxes the last week of Feb to take advantage of Turbo Taxes discounted pricing but not pay the taxes owed until April 15th. I will have the money by then.

Yes, when filing now you can set the payment for your tax for any date up to and including April 15th.  You can also indicate in TurboTax that you are "paying by check" (whether you intend to mail a check or pay through a different method) and then schedule or pay it whenever it is convenient directly with the IRS.