If I use after-tax dollars to fund an HSA, do I get the tax back for those dollars? Federal and State? Does this require a different form?

You get a tax deduction that reduces your federal taxable income, which should also flow through to the state taxable income.  In Turbotax, your employer/employee contributions are captured from your W-2.  Later in the HSA interview, you are asked if you made additional after-tax contributions.  The employer/employee contributions go on line 9 of form 8889 and the after-tax contributions go on line 1 of the same form.  As long as your total contributions are equal or less than your annual limit, and you are enrolled in a qualified insurance plan, you will get a tax deduction for the after tax contributions.