Is it permissible for me to live in the female student dormitory near the university without a mahram, knowing that I can move to another state with one of my mahrams after one academic year?

You have done well by leaving the mixed-gender university and moving to one where there is no mixing. If the new university requires travel, then travel is not permissible except with a mahram. As for residence, it is not obligatory for it to be with a mahram; rather, what is obligatory is for the girl to be in a safe place for herself, such as a dormitory for female students.