I agree.. don't quit your job. Also, as said above, $15 an hour jobs do not grow on trees. I was unemployed 7 years ago and after 5 weeks considered myself very lucky to get a job at $10 an hour with NO benefits.
Getting another job in the fall will be problematic. Trust me.
As to kids at 12 and 10 being home alone, yeah.. they can be. Pay them an allowance or a wage to be responsible and give them a list of things that they need to do every day. You will get things done around the house and they will learn about earning money.
At age 12 I was home alone, my younger brother was 9. I had chores to do and so did my Brother. Fact is, at age 9 he was restoring Antique Victrolas and I was training horses... We would sometimes take the horse and ride double down to the creek and go swimming.. or take the horse and go try to catch Painted turtles (don't ask.. LOL). When my Brother turned age 10 he was cleaning houses and painting and doing yard work for ppl to make extra money.. it wasn't so bad.
Whatever you do, do NOT quit your job. That won't look good at your next job interview. I learned the hard way.. NEVER leave a job without having another one to go to.