If you've held the stock for more at least one year, you are correct, and you will be able to deduct the Fair Market Value of the donation.
However, if you have held the stock for less than one year, it is short term capital gains property, and you can only claim your basis as a deduction. See page 11 of this link: https://www.irs.gov/pub/irs-pdf/p526.pdf.
As long as you've held the property for one year, the amount that you enter for the basis will not affect your deduction.
If you think you are not getting the FMV, it might be a misunderstanding or it might be a bug. You will need to look at your schedule A totals before you file. Be sure to review everything carefully before transmitting your return in case something needs to be fixed. In Turbotax online, you can't print your actual tax forms until you pay the fee, usually at time of filing, but you can pay early, print your return in progress, and still go back to the interview to make further changes.