Is it permissible to perform tayammum for prayer when clean water is readily available nearby, but there is a legitimate religious impediment to using it, such as the fear of exposing one's 'awrah in front of strangers? And is it valid to suffice with a shower (dush) instead of ablution for prayer?

Ablution (Wudu') is a prerequisite for the validity of prayer (Salat). Dry ablution (Tayammum) is only permissible when water is unavailable or one is unable to use it. Therefore, the prayer that was performed with Tayammum, while it was possible to perform Wudu' from the sea, must be repeated. As for the husband's ritual bath (Ghusl): if it was an obligatory Ghusl, it suffices for Wudu'. However, if it was a recommended or permissible Ghusl, it does not suffice unless the proper sequence was observed with the intention of Wudu'.