What should I do if the time for prayer arrives and I am in a gathering where I cannot go to the restroom for purification and ablution without being put in an embarrassing situation due to the proximity of the place and the audibility of sounds? Should I pray with only ablution and then repeat the prayer later, or what?

If you are not in a state of needing to relieve yourself when attending prayer, it is permissible for you to perform ablution and pray, and you are not required to repeat it. However, if you are in a state of needing to relieve yourself, you must first relieve yourself, then perform ablution. If you pray while needing to relieve yourself, your prayer is valid, even if it is deficient. If the urge to relieve oneself occurs during prayer but does not prevent its completion, the prayer is valid without being disliked. It is obligatory for those who hear the call to attend congregational prayer, and it is not permissible to delay the prayer beyond its time for the excuse of relieving oneself unless one fears that the time will expire; in which case, one must relieve oneself and then pray. However, if there is ample time, it is permissible to delay the prayer to relieve oneself, as long as the time does not expire or the congregation is not missed.