Texas homestead exemptions are filed on a primary residence with the appraisal district. The general residence homestead rules are in Tax Code Chapter 11. A cap on how fast the appraised value of a qualified homestead can rise is in Tax Code Section 23.23. The cap does not work the same way in the year you buy, and an exemption you file after closing does not rewrite the estimate the lender used on the Closing Disclosure.
At closing, escrow is an estimate from the tax bill or the title company's tax certificate. If the house was previously taxed as vacant land, or the builder's estimate was low, the next assessment can be higher. The payment changes because taxes changed, not because the note rate changed.
School taxes are a large part of a Fort Bend or Katy bill. Confirm the school district on the CAD account before you rely on a neighbor's tax figure. Fort Bend ISD and Katy ISD do not charge the same rate.