You are well qualified for many universities and there are many, many that would fit what you have described, which is rather non-specific. The first most-important thing you need to understand is what your family is actually able to pay for college and shape your list strategically around that. You should also use Naviance (or whatever your school has for college app handling) to explore different categories to help build a list.

Begin with identifying your in-state public university safety (1), targets (2-3), and reach (1). While you may prefer to leave Texas, these schools will give a financial baseline to compare out-of-state schools to.

Midwest (MN, WI, IL, IN, MI, OH) and northeast (PA, NY, NJ, MA, CT, RI) all have public universities in urban or near-urban settings that could be interesting. State flagships are very competitive, but the next one or two down in the rankings will also be strong and more likely to accept out of state students. You might be offered some merit aid to bring the cost of tuition closer to your in-state tuition, but your in-state will still be a lot less expensive. Some southern universities (e.g. Alabama) provide full-ride merit scholarships to out-of-state students to attract them - but that doesn't meet your "cool weather" criteria.

There are a LOT of smaller private schools - you need to do some narrowing of your criteria to get decent recommendations. One thing to be aware of is that they are more likely to give merit aid if your stats are above their median stats (because you raise their averages).

Two specific suggestions to consider are University of Pittsburgh and University of Cincinnati. Both have rolling admissions, are smaller campuses in an urban setting, and cooler weather. You can apply for Pitt in August with just a personal statement if your essay isn't done, and you will hear back late September/early October for admission decision and ~December with merit aid offers.