Understand Houston does not have a "nice" side of town and a "bad" side of town. There are no city zoning laws. You can be in an expensive addition, make a wrong turn and go a mile or two and be in places you shouldn't go after dark. Houston goes from good pockets to bad pockets and then back to good again as you drive. The immediate area around U of H and Texas Southern ARE bad; some areas are really bad, even dangerous. However, River Oaks - one of the most expensive additions in Houston which has old true "estates", is only 7 miles and about a 15-minute commute away from U of H. Minute Maid Park and the George Brown Convention center downtown area, very expensive real estate with million-dollar condos, are just blocks away. The University Area addition around Rice University is some VERY expensive real estate, is only about 3-4 miles away from U of H (homes in the $1-20 million range). All nice places to live, all more expensive than anything you have in Stillwater.
Alternatives would be to go South down I45 to some of the nice suburb additions there, but you are talking an hour or more commute each way in stop and go traffic. To live in a nice addition within 20 minutes or so from U of H you are literally going to spend a fortune.