I was unemployed for couple of months and during that time I interviewed with two companies. I asked to both a salary of 65k$. I like ways more the second one, but the first one was faster and they offered me a contract with a salary of 45k$. They were not my favorite company and I thought that I could have the other offer,so I asked more money and some days to think about that, but they said "no way, if you don't like the pay we go for someone else". Having to pay different things, I signed with them. After 3 days the second company showed up and offered me 60k + some bonus. Everything is better with them. My contract says that I can quit the job with two weeks period notice in the first month, would be fair to start the job on Monday and quit the job on Tuesday? Or this is not fair? What would you do if you were me?
Have you started yet? If you haven't, just give you notice immediately. If they want to go for somebody else, now is a good time for them to do it. If they're rational, they'll let go of you immediately rather than pay you for two weeks and end up having to get someone else.
Such things can be legally binding, and tricky to navigate out of if you don't have proper documentation. Email the person who employed you, CC HR, and state that you would like to come in for a meeting- bring your formal letter of resignation and list your grievances. It is not wrong for you to not want to work somewhere where you are not valued for the skills you bring to the table.
Best of luck in this process, and with your new job. Remember, the decision affects you (your family) so don't hesitate to speak up for what you want!