Ultimately, there are two issues that come into play:

1) Our state government is often at odds with the cities on this. Case in point, the Lieutenant Governor and the state legislature nearly passing a bathroom bill in 2017. It was only stopped because the Speaker of the House put his foot down and refused to allow a vote. Businesses were vehemently opposed to a bathroom bill, but that didn't matter to the Lt. Gov, the state Senate, and all the members of the House who would've voted for it (it would have been a majority) if the Speaker allowed it. The extreme conservatism of the state government also means that there's no statewide non-discrimination laws (and the legislature came within a hair of revoking the ability of cities to have their own non-discrimination ordinances in 2017, but Speaker Straus squashed it along with the bathroom bill), which means that if you're openly LGBT and you set foot outside one of the little urban islands of blue, you're basically screwed. The state government will also regularly target blue cities' regulations, as well. In 2017, the state legislature revoked the cities' ability to regulate rideshare companies in response to Austin passing consumer-protection laws that Uber and Lyft didn't like. More recently, the state AG is suing Austin for passing a mandatory sick-leave law arguing it violates free enterprise. If you're a blue person living in a blue city in a red state, this is what you'll have to deal with.

2) King of the Hill is never far away, no matter how blue your city or neighborhood is. If you're in Deep Ellum, a very blue hipstery neighborhood of Dallas, you're only a stone's throw away from Garland, which King of the Hill was actually based on. If you live in deep-blue Austin, you're not far from Pflugerville and Round Rock, which are much more conservative. Now, personally I don't mind it. As liberal as I am, I live in the suburbs, and I love living in the suburbs, but well, you get the confluence of both this point and the above one...

Keep in mind the way our cities are laid out, there are seamless transitions between cities and suburbs, and our embrace of edge cities means that much of the local economy is in the suburbs. People who live in blue cities may very well end up spending most of their time in red suburbs. This means that a) the lack of statewide discrimination laws and other protections will affect you personally, and b) people who don't want to set foot in Arlen will often end up having to do so. You can't just say "I'm going to move to Deep Ellum and never set foot outside the neighborhood" because the metro area isn't set up for people to get everything they need from one neighborhood or even from one city. You can expect to have to set foot in some deep-red parts of town for work or shopping or some other essential service.

For the record, I'm a liberal LGBT Texas native who has never moved out of town and currently lives in a conservative part of Dallas proper and works in a very conservative suburb. I've only ever considered moving out of town when a bathroom bill was on the table, and the Lieutenant Governor's recent announcement that he's given up and isn't going to pursue it this year has made me decide against a move I was considering. I love Dallas to death, and I never really wanted to move even in 2017 when I was worrying myself half to death over the bathroom bill (I would have moved if it passed, but I would have seen myself as a refugee, and I'd never stop pining for home), but it's despite the state government's shit, and I can understand why a lot of people just don't want to deal with it at all.