Is it permissible for a girl to propose marriage to a man she deems righteous, especially if she doesn't see him often and feels shy around him?

A woman is permitted to propose marriage to a young man if he is pleasing in his faith and character, while adhering to Islamic etiquette and avoiding private seclusion (khalwa) and speaking in a submissive tone. If he desires, let him propose to you through your guardian. We advise you to perform Istikhara (prayer for guidance), and if the matter is facilitated, then all praise is due to Allah. Otherwise, ask your Lord to facilitate something better for you, for supplication is the weapon of the believer.