You should be entitled to Unemployment Insurance benefits. Make a claim! You pay into it from every paycheck, so it's yours for exactly this circumstance! Your employer has to forward a Record of Employment. If you had a high salary, there is a top-out but the monthly benefit should be decent (and beats nothing at all). This pretty much involves taking the severance, but you can ask for more severance. Here, severance is one month's pay per year worked- so you would get 8 months severance. I take it this is not what you were offered. You can ask for more - it can't hurt. Call before your deadline is up. You also might want to consult a lawyer without committing to go to court, to know your rights in your area. If you know anyone who knows law, or who has been laid off themselves, they would be a great free resource.
Remember a lot of your expenses fall away when you're not working: commuting, lunches/coffees/etc. Your friends will be sympathetic. Also - you have plenty of "stuff" to hold you over until you have a job, so you don't need to buy anything new (or used). Coupons aren't really your friend - they get you a slight discount on expensive products. Learn to cook for yourself, and use less of things you can't make. This will do you more good than coupons.
Do what you have to to help your situation. Can you move houses, or sell some of your things (you can always buy them again when you get a job). Look around, even a few things worth $50 would be worth throwing up on Craigslist. If you have something valuable, you should definitely consider selling it.
Consider in all this: what length of unemployment do you expect? Are there a lot of people unemployed in your area? Would you consider another field of work? I highly recommend going to an employment agency - they have access to a lot of great jobs, and on the job-seeking end you don't pay anything. Temporary work can mess with your employment insurance benefits, but they often recruit even for high-level full-time jobs.
Lastly: have FUN! You have less money, but you have TIME! Do dedicate some time each week to job searching, but now is the time to do things like hike, hang out in spots you normally don't get to, etc. I was lucky to be unemployed in summer, but set it up so you're not too worried about money and take pleasure in long walks, bike rides, sitting in a park, or whatever pleases you. Sleep in! Clean your house (you'll have more energy for it!). Go see your friends (at their home) on a week night if they'll have you - you can be up late. Do lots of free activities with your kid (and remember this is the #1 thing kids need and want in life! You are now a stay-at-home parent - something many people dream of.) Personally, I also did a bit of volunteer work because I actually had the time for it - and I would rather have that lifestyle than full-time work any day! You might be in this for a long time, so build a great life around unemployment and seize the opportunity for free time given to you. You do yourself no favours by being mopey and downtrodden, and actually you'll come across better in interviews. When they ask why you left your job, you can even say something like "I wanted to spend more time with my kid and do some volunteer work, so I have taken time away from work to do it, and now I'm looking for a new job". When they ask why you're switching fields, you can also come up with something positive - as long as you yourself have a positive demeanour.
Best of luck! It's tough but if your first step was r/frugal I'm sure you'll be fine in the end!