Call the title insurance underwriter, not the title company. Look at the first few pages of the title commitment for the numbers or the last few pages after schedule D. Policy should either pay it or give you reason why in short order.
Pay your current year's taxes on time. I know, this is after July 1st.
Calling the assessor's office really doesn't get you anywhere because the ones you are talking to are not lawyers or have any authority to change (and zero motivation to do so).
See Property Tax Code, Sec. 32.01. TAX LIEN and Sec. 32.07. PERSONAL LIABILITY FOR TAX specifically. See also, Sec. 25.16. PROPERTY LOSING EXEMPTION DURING TAX YEAR.
Tax liability attaches to the owner of record on January 1st. If an exemption is lost, it attaches to the owner of record as of January 1st of the year for which the exemption no longer applies. In short, the tax bill should go to the seller for those years they owned it (or their heirs, if that's the case).