What should I do, as I vowed to wear the niqab upon memorizing the Qur’an, but I found that I can no longer enter public places because of it?

Wearing the hijab and niqab is an obligation upon every Muslim woman. Your vow to wear the niqab once you memorize the Qur'an is invalid, because vowing to perform an obligation is not binding according to the majority of scholars.

If wearing the niqab leads to significant harm to oneself, or to one's limbs, honor, or mind, then it is permissible to uncover what wards off the harm, while adhering to regulations such as not going out except for necessity. However, if the harm can be endured, then she must be patient and steadfast, and she will be rewarded for it. If it is possible to move to another country, then you must do so as a flight for your religion.