Everything is up to negotiation. You do have leverage, your lawyer will tell you how much.
The process would be:
1. Don't sign anything.
2. You hire a lawyer who specializes in employment disputes. If the severance package has a deadline, your lawyer will email the company saying they represent you and need more time to respond.
3. You meet with your lawyer and explain your situation. My lawyer explained to me that when you get laid off, your company is breaking your employment contract. That means they have to provide you with compensation to make you whole. The amount of compensation depends on your skills, experience, job level, age, health of the job market and any other specific circumstances. Basically, the company has to compensate you enough money that would make you whole - which is be employed again. The compensation is based on the time it will take you to get a new job. 3 weeks is kinda short. Also, there maybe rules-of-thumb: in my city in Canada, we expect to get a minimum of 1 month of salary for each year you worked with the company. I have seen some very generous companies give 2 months of salary per year worked.
4. The lawyer will write up a demand letter asking for more severance. From that point on the lawyer does all the communications with the company. My company also offered employment services to help me find a new job which i didn't need, so my lawyer said ask the company to pay that to me in cash... was like $5K. My lawyer said there is zero risk in sending a demand letter - never will get a worse package.
5. Depending on the size of the company, they may expect that a certain percentage of employees will dispute their severance package - which means they already planned to give complainers a bit more... as in, already authorized, you ask, they give... easy.
6. My company said "No" to the demand letter. They pointed out that there were many jobs that I would be hired for and my severance is enough. They provided the job postings that I would be suitable for.
7. In response, my lawyer told me to go through each job posting and write a couple sentences of why I am not qualified for them. My lawyer sent that to the company.
8. Company responded with a higher severance, an extra few months of salary on top of the 1 month per year and the $5K mentioned above. The legal costs were less than $2K.