In the Chapter of Repentance, God Most High reminds the believers that He is most deserving of their fear, “Do you hold them [the disbelievers] in awe? Nay, it is God alone of whom you ought to stand in awe, if you are [truly] believers!” (Al-Qur’an, 9:13)
The lesson from this verse is important. No one is more deserving of our consideration than Allah Most High. People are people and they will have to stand before Allah just like you will; in other words, we’re all nothing before Allah. Why be uncomfortable if someone sees you performing ablution? Public facilities are just that. Public. No one can stop you from washing yourself. And if you start performing tayammum (dry ablution), not only is it invalid since you have access to water, you’re depriving people of the opportunity to interact with you and learn something about Islam.
Finally, we live in a society where people have very little public shame. So we need to check ourselves to see why we put so much stock in what random strangers think of us. Who cares if someone has a problem with our hijab, beard, or foot-washing? We have to be confident in who we are as believing men and women, aspiring only for the approval of our Lord.
May Allah Ta’ala grant ease