It has been thirty years since I've graduated (RN). I am ever so grateful that I had the opportunity to have a supportive management and orientation for my first job. This was at a mjaor medical center in NYC. With that said, they ran us ragged too, but at least there was support. Yes, a mistake was a mistake, but we dealt with it in a positive way.Had everyone been fired for a mistake (or lack of working equipment), there would have been no one left to care for patients.
With that said, try to find the very very best place to work . Don't try to find the "first job" that will "take" you. As a new nurse you want the best in mentorship.
Do you still have a good relationship with anyone ( a former professor) at your nursing school? They may be able to help you through this time. I hate, hate seeing this type of thing happening to decent folk. But frankly, I would stay away from LTC. Get yourself into a teaching hospital with decent mentorship and leadership.