I used to live in a fairly nice place called Trails of Woodlake that was near Westheimer and Gessner. I would guess a 1-bedroom would be $650-700. Surrounded by woods, mostly decent neighbors. Some of the apartments were rented out and some were owned. I can't tell you much about the management since I rented from an owner. They were alright, though they flew the Texas flag upside down. Pretty safe area. Had a courtesy officer at the gate though I can't say it was an effective system. Mostly, it was protected by its geography and tenant profile.
BTW Mulder, these aren't efficiency prices. The $850 apt. is still humongous for a one bedroom. I know that's still $200 more than you want to spend, but it might be worth looking at, unless that is still too high.
Mulder and drapg probably used to be my neighbors... not a bad area to live in, and close to a lot of stuff, but still crowded. And I don't even like having to squeeze two people into a one-bedroom apartment. (of course, now that I live and work in the middle of nowhere, I rent a very nice house, in good repair, with 3 bedrooms, for only $800. )
