Please recommend me with good schools for construction management major

You could take ACT if you can score better on that. Deadlines are coming up for applying for merit for fall 2016 admission.

Some civil eng depts have separate structural concentration. Auburn has construction managment; DD is at UA (University of Alabama) - look up how diversified civil eng is there, with four areas of concentration. DD is taking extra electives to have concentration in structural and the general civil eng.

If your ACT is high enough, you could qualify for tuition scholarship and eng scholarship. Also if your M + CR score. Automatic scholarship. Look at www.ua.edu for details.