I don’t think the online backlash matters. The real question is if Abbott is receiving any backlash in Texas and it appears probably not much, if any. Texans need to step up here and they don’t seem to be doing that.
Blue areas tend to be disgusted with his actions, red don't care at best or think, "well that's what they get trying to cross illegally". The problem however, is that the districts are gerrymandered like crazy and the actual polling places tend to disproportionately be in Republican areas, which makes it hard for Democrats to get a real hold anywhere they don't already have control of. Additionally, the Democrats really feel like they aren't sending their best. Iirc Beto didn't originally want to run for governor, but he did because all the other options were so bad that very few people even knew who they were.