36 Years Old Mid-Career with No Savings or Retirement

Made a new account to post this just because it's embarrassing. Looking at some post here, I can't believe how much some people have saved. I'd like to make this the year I start correcting the second part of the title and figure out how to really handle my money.
 
I'm married with four kids (adopted). I've been contracting the last few years in the product space and make okay money (~$150k) for living in a low cost of living area (Arkansas, US). I'm the only one working in the home partially due all of the kids having some kind of special needs (nothing ultra severe, but so many appointments). I have a mortgage for our house, two fairly low car payments (although one might be about to become a bit higher if we trade in for a car capable of carrying us all semi-comfortably.) Kids are all on Medicaid, so that helps.
 
All in all, I never really feel like we're doing poorly, except that we tear through our funds every week and every time I whip out Excel and try to make a budget plan, I end up looking at a bunch of months with negative money leftover and no personal spending or savings.
 
I'm not really sure where to go or what to do. I've been cutting what I can, but I'm running out of fat to trim. We have no credit cards with anything on them, no student loans, I think a pretty reasonable mortgage and car payments (let's assume I don't get the minivan). What's killing me here? How are other people saving so well? Is there a recommended budget program? I hated You Need a Budget after it went to a subscription model and Excel is... prone to my user errors.
 
Apologies for the rambling. Any help is appreciated. I've been going through the sidebar and will continue to read through.