Narrated Abu Sa`id: The Prophet was more shy (haya) than a veiled virgin girl.
What other sahih hadith are there that specifically describe The Prophet (peace and blessings be upon him) and his sense of haya.
(For example: I have heard narrations like “The Prophet's face (piece and blessings be upon him) would immediately change (turn to red) if he was exposed to even the least bit of anything that would be considered unshameful.” But my search brought nothing.)
It seems to me you refer to the sahih hadith:
The Prophet (ﷺ) was more shy than a virgin in her separate room. And if he saw a thing which he disliked, we would recognize that (feeling) in his face. (Sahih al-Bukhari and Sahih Muslim)
Mullah 'Ali al-Qari commented the statement (if he saw a thing which he disliked) in his Mirqat al-Mafateeh sahrh Mishkat al-Massabih مرقاة المفاتيح شرح مشكاة المصابيح -see here in Arabic- as follows:
In the following I'm translating from Arabic as these are my own translations take them with the necessary.
فإذا رأى شيئا يكرهه ) أي : من جهة الطبع أو من طريق الشرع This means either due to his moral constitution or due to the rulings of shari'a.
He then goes on:
( عرفناه في وجهه ) . أي من أثر التغير فأزلناه ، فإنه ما كان يعاين أحدا بخصوصه في أمر الكراهة دون الحرمة . قال النووي : معناه أنه - صلى الله عليه وسلم - لم يتكلم بالشيء الذي يكره لحيائه بل يتغير وجهه ، فنفهم كراهيته
(we would recognize that in his face) from the clear signs of change so that we would change/remove it, as he was not used to survey a single person in case of dislike which doesn't reach haram. An-Nawawi said: it means that he -Allahs prayers and blessings be upon him- was not used to speak about something he disliked due to his shyness (haya') but his face changed so that people understood or concluded his dislike.
