You will only be able to use your state of residence (your home of record - TX) when taking the sales tax deduction on your income tax return. Since the car was purchased in VA, you will not be able to include this car purchase as a major purchase under the sales tax deduction section.
Follow-on related question, I am active duty military stationed overseas (Germany), should I use my state of residence (FL) for the sales tax deduction? If not, how can I remove the sales tax deduction after it has been applied to my federal income tax return?
Yes, use FL for calculating your sales tax deduction. Using FL as your basis for calculating a U.S. sales tax deduction is consistent with the intent of the U.S. tax code.
As an active-duty military member, your income is treated as earned in FL and you should treat any income-related deductions (i.e. sales tax) as if they were also sourced in FL.
The IRS doesn't have any way of calculating the U.S. equivalent of the German sales tax (VAT) - their tax rates and currency cannot be easily translated to a U.S. equivalent sales tax.