It really depends on what your expectations are for that budget. We are in Creekside in The Woodlands and love it here. We are in Tomball ISD, but I would not call where we live Tomball. The school district lines are just crazy. Your taxes and insurance will be lower if you buy in Montgomery County, though. I've been happy with our elem school because it's brand new and was built in the subdivision. We can walk there, if we need to. It's also K-6, which I prefer. A brand new junior high is being built across the street from our elem school, and it should be open by the time my kid gets to 7th grade. I'm not thrilled with the high school, mainly because it's so far away, but I have a while before I have to worry about that. Having good schools right near our house was a big priority for me.
With that budget in Creekside, you could probably get a resale with about 2500-2800 sq ft that is less than 5 years old (Not new construction, though. Those prices in Creekside have gone through the roof). Options in Tomball, and Spring/Klein should be a little cheaper. I've heard great things about Klein schools as well. You just have to be careful if you move to a more "rural" area with schools that zoned for larger areas with a less dense population.
I would expect that when Woodforest is built out, that new elem school will service kids that almost exclusively live in Woodforest. That's what has happened in Creekside. One of the two elem schools has only kids who live in the Creekside development. The other one has a few from outside the subdivision, but not too many. Creekside is big enough that, in addition to the junior high, they have also started construction on a third elem school. There are pros and cons to having a school that serves such a limited geographic location, but so far, the pros for us have greatly outweighed the cons.
I'm not too familiar with the rest of Tomball, just because I didn't really look in that area. My search was limited to Montgomery County, and then Creekside, when I couldn't find anything I liked in MoCo.