I was born in Dallas and have lived in the area all of my life. I have personally lived in several areas of Dallas, Carrollton, Farmers Branch, Euless, North Richland Hills, Grapevine, Hurst and Colleyville. If I had my choice of places to live in the metroplex, it would be Grapevine. To me it has a down-home feel with all the amenities of a metropolitan area.
In regards to specific streets and neighborhoods, it's hard for me to say. I lived in 2 different neighborhoods in Grapevine, both of which worked perfect for us at the time, but the houses in both of those neighborhoods are selling for a lot less than your current budget (smaller and older than what you probably need). If you do a search of Grapevine (zip 76051), in your price range, you'll be able to see what you can get in that area. I will say most Grapevine home values are higher than some other areas because it is very popular, the school district is highly rated, and the land is pretty much all built out so there are very few new homes going up.
You can get cheaper homes with more yard in areas like the mid-cities (Hurst-Euless-Bedford) and Keller, but they are a bit further from Las Colinas. The other two nicer areas around here are Colleyville and Southlake, but most of the neighborhoods in those towns are pretty high end, so homes in your price range might be much smaller and less desirable, although, I've seen some decent things come up in both Colleyville and Southlake recently at reduced prices.
I know the areas I'm talking about are all pretty much west of Dallas proper, but it's the area I know best. My mother lived in Dallas until about 6 years ago, but I wouldn't recommend her old neighborhood now as it's going downhill daily. She lives in Grapevine now though. :-)
If you find any homes you're interested in in any of the neighborhoods I mentioned, please feel free to PM me and I'll share whatever information I know about the specific area you're looking in.
Good luck! I hope you are able to find something you love very soon!
Blessings,
N. :-)