HB 39 is a bill that actually bans employee vaccine mandates (unlike Abbott's half ass order that only bans it for state employees). Call your rep to push them to support this or you will do everything to primary their ass.
https://capitol.texas.gov/Search/DocViewer.aspx?ID=872HB000391B&QueryText=%22HB+39%22&DocType=B
https://capitol.texas.gov/billlookup/BillStages.aspx?LegSess=872&Bill=HB39
They have had three legislative sessions in 2021. 87(R), 87(1), and 87(2). R for regular and 1 & 2 are emergency/extended sessions. We are in 87(2) right now. The bill you linked was an earlier session. The one I posted originally is pretty new and in current session.