Your Plan is pretty good.
1) Pay off the debt (except mortgage)
2) Based on your budget, I think 3 months of expenses would be about $9,000. You have a one income family, with multiple children, and a history of going in to debt for either emergencies, wants, or needs. I think you need more in an emergency fund than $9,000 because of that. But, $9,000 should immediately go into a savings account at an online only bank where it would take you a few days to access the money.
3) Spending $1,000 on a new CPU/ tower for the computer and a new laptop combined seems like an appropriate splurge. A $300 laptop should be perfectly sufficient, perhaps a Chromebook? I think a $400 computer would be good, and then a $150 screen.
This leaves you with $10,000. I'd spend the money on the house beams before the education. Then, if the money will help you refinance the mortgage to a lower rate (either the same payment/shorter term, or longer term lower payment) I'd do that next. Then, I'd save the rest either for a replacement car when that comes up, other house emergencies, or needs you aren't seeing right now.
Then, you need to figure out your spending. $850 on food/ Costco is pretty high for a family of 5. Do you think you could get it down to $600 a month total? Your utilities also seem pretty high to me. Cut the landline.
Will your health insurance go down with an Obamacare subsidy? Maybe you can cut $50 a month off of that somehow? $150 a month seems high for car maintenance, do you really use $1,800 annually? Do you think you can, with these means (insurance, food, healthcare, landline, mortgage refinance), cut your budget $200 a month? Then, without the debt, you'd also cut your budget an additional $468 a month. "Pay yourself first" with this $668, into the savings account until you can afford the 30+ classes from new savings not from the emergency fund.
You say you have nothing left at the end of the pay period, but it isn't going to luxuries. You need to figure that stuff out and budget for it. Is it: Personal/medical care (haircuts, contact solution, pharmacy expenses, co-pays, dental, underwear/socks), Restaurants/ fast food? Gifts/parties/ activities for the kids and their friends? Gifts for adult friends and family? School supplies/ curriculum? It is probably all of these things over the course of a year.