I don't live in Ft. Worth now, but I grew up there, and my dad still lives there, so I go back a fair amount.
I'd say, as far as ambiance, what you're looking for is likely in the Cultural District or Sundance Square, both very chill and very walkable areas in downtown Ft. Worth, near where you work. (The museum area is the "Cultural District," and Sundance Square isn't far and has more amenities, like grocery stores.)
However, as you've already noticed, both of those areas are the most expensive parts of Ft. Worth. It might be worth it to you, though, because you could bike into work easily from the Cultural District and walk from Sundance Square, and still feel like you could go out at night and not be as isolated as you might be in Arlington or Mansfield.
Having said that, Arlington is generally a pretty nice town, no need to be worried about neighborhoods. (As is Mansfield, but I didn't see you express the same reservations.) Both will be much more inexpensive than Ft. Worth, but (as has been noted) you'll have to contend with a commute. If it's possible, going to work at slightly "off" hours (i.e., coming in at 6:30 or 7am and getting out early) can help with that. And the good news is that your office building is close to public transportation.
The other "negative" of Arlington/Mansfield is that people tend to be more settled in those areas. That is, people who live there tend to have lived there for most of their lives. And there's a very high proportion of families in those places. You don't mention a family, just a dog, and so for singles who might be a bit newer (and possibly more open to making new friendships?), the downtown area could be better in that regard.
Let me know if you have other questions. Hope that helps!
