Question

In a verbal agreement, until when can a person back out, and from what point is the matter considered set? And if you back out after that, is that considered unreliability, irresponsibility, and a bit immoral? 
 
For example: I had to stay overnight at a hospital on Passover. The only solution was to switch with someone else. I asked two people. The first answered yes, but for one hundred shekels (for example), and the second answered yes, but for one hundred fifty shekels. They both wanted money, because if they switched with me they would lose a workday at their jobs, so I had to compensate them for that.
So I agreed verbally with the first one and told him, “Okay.” Five minutes later, the second one got back to me and said, “You know what, fine, I’ll do it for free, because my religiosity (Islam) doesn’t allow me to make money off other people’s problems.”
So I backed out and told the first one to forget it, since I had found someone who would do it for free.
Did I act properly?