I can 100% assure you, that not a single General or Commander for that matter, had been exempt of "losing face" as you describe it. It is common, and it's almost an untold requirement to climb up ranks. The only difference in what makes a good leader and what doesn't, is the way you handle such key moments.
You have the ability to turn something bad, into something helpful for you, something admiring, and turn the public spotlight to let everyone know you have an admirable way of handling any type of situation, and only then, you will gain the trust from even those that you were expecting to do the opposite.
Always remember, possibilities are always in your hands, and you came to a place where you have exceptionally understanding leaders, so, my advice is for you to take good advantage from it, and turn a moment of shame, into a moment of respect.