Severance isn't something that's guaranteed in most places. So there really is no normal or standard amount. Most give nothing. Some give a week or two. I've seen some companies offer something like 2 weeks of pay for every year of service. So it varies. If your company is offering 1 week, then it's one week.

So when you say you will agree to not go after more than you're entitled to.....you're unfortunately not entitled to anything. And if they're firing you, you probably have no leverage here. Review your company handbook/policies to make sure they are complying with any documented termination policies. I guess you could consult a lawyer if you think they may have violated some local law in firing you. But beyond that, I'm not sure what leverage you'd have to ask for more.