I don't think Stannis would've killed her. I just think she recognized that Stannis was a lost cause.
In terms of the losing faith thing, I think it's more that she's questioning herself as a priestess and her own abilities, given that she interpreted things a particular way, and they didn't turn out as she had interpreted them. In the books, as mentioned, she has a POV chapter or two where you see that she kinda makes it up as she goes. I think they did this on the show, too, back in season 2 or 3. Anyway, she makes it clear that the certainty with which she speaks is kind of a front. She's sure that Rhllor is guiding her to Azhor Ahai, but she pushes her interpretation with Stannis.
One of the things mentioned in the books -- and I think also in the show -- is the unreliability of loosely worded prophecies and how adherence to them tends to bite people in the ass. When she said she saw Bolton banners in flames...I mean, yeah, maybe she did. Doesn't mean STANNIS set the fires. Maybe they burn because the candle that Sansa lit falls over. Maybe they burn because Jon rises from the dead and burns them. Maybe they burn a decade later when Dany comes to Westeros with full-grown dragons and torches 'em. But her interpretation was "Oh, my king, you will definitely win. I have seen it in the flames."