As others have said, when you're in a financial bind, your only options are to reduce out-go or increase income. Except for a couple of suggestions (below), it sounds to me like you are maxed out on the income side of the equation. I strongly suggest you take another look at the out-go.
Let me put it this way. You and your wife are struggling. Your days are filled working, taking care of the children, taking care of the house and paying bills. There isn't a ton of joy in your life because there simply isn't any time. For you or your wife to work more means neither of you are even going to see each other. That's not good. Your job already has a high risk for alcoholism, drug abuse and divorce. Let's not put more stress on you and the family.
My suggestion is to stop the cycle of stress. It is going to take some hard decisions that both of you will need to make. It takes commitment. Working more, working harder is not going to save you. Making a decision to live differently IS.
Suggestions:
* Sit down with your wife and TALK. Talk about your values, priorities and goals. What do you each, individually value the most? (there are list of 'value' words on the internet that you can start as a reference point.) Compare your lists and create a combined family values list of no more than 5 (preferably 3) values. Prioritize everything your family does around these core values. Make every decision based on these core values and how they are prioritized against each other. Then compare how you're living now to these values and if every moment of the day isn't devoted to living these values, write goals to get your family to that point. DO NOT expect you'll be able to do this all in one discussion. Writing family values, priorities and goals for my family took months.
* Search out some books for inspiration. My favorite, the one that turned my life around, is "The Tightwad Gazette" by Amy Dacyczyn. Other people swear by "Your Money or Your Life". Other older classics are "Think and Grow Rich" or "The Richest Man in Babylon". On this board people love Dave Ramsey and you might consider doing his Financial Peace course. An on-line version is available that you can work around your schedule.
* Track every single penny. I can't stress this enough. Account for every penny coming into the house and every penny leaving. EVERY PENNY! Yes, write it down. You're not just paying bills anymore. You are now managing your money. Make managing your money your new part-time job. Whatever time you were going to devote to working another job, devote to managing your household finances. So many people put money on auto-drive and steer right off the road. You, instead, are getting behind the steering wheel and driving this thing.
* When reviewing your spending, consider addressing categories that we all assume are fixed expenses but really aren't. A good example is taxes. If you get a big refund year after year, adjust your withholding. You need the money now, not later. (This is also why a part time job never works out, the extra income almost always gets eaten up in taxes.) Look at insurance premiums, deductibles, discounts on utilities (they are available!) Call all your providers and say, "I need to lower this bill. How can I do it?" When looking at taxes, take into account you have this huge medical bill which might be a deduction.
* Look at your spending on consumables - stuff you're buying only to throw away or wash down the drain. This includes: paper towels, toilet paper, paper plates, plastic baggies, aluminum foil, newspapers, magazines, trash bags, toiletries, make-up, cleaning products, etc. Many people try to reduce their budget by cutting back on food. Instead I suggest the first place to cut back is non-food consumables. For instance, in your line of work, you many need to buy a heavy-duty detergent but the whole house doesn't need to wash their clothes with that same expensive stuff. Some loads could be cleaned with $2.99 Purex detergent and some loads like bath towels might get perfectly clean if they are washed once in a while with nothing added at all. There is often so much soap residue in some fabrics, they could do with a washing in plain water.
When my husband got laid off I stopped buying all consumables except TP (the WalMart version of Scott 1000) and garbage bags (because our trash service requires all garbage to be bagged). For 14 months, I made do with everything else we still had in the house. When he landed a job, I ran out and bought a roll of paper towels. Oh! The luxury!
When we were newly married we had a situation where we had to help family and suddenly we had an extra $800 a month financial obligation. We were already more-than-full-time working professionals with a house and bills and all that so I really didn't know how we were going to come up with that $800 month after month. But we did it! We used the pantry principal. We refused to buy anything, especially clothing if it cost more than $5 (your work clothes are exempt because they are tax deductible). It was a rough couple of years but it proved to me we could live on much less than we thought. BTW, during that time we went through a major natural disaster (and recovered) and traveled. We weren't just living on peas and neck-bones.
* Take advantage of government services if you qualify. DO NOT assume you make too much. Unless you make over $90K a year and have one child, you might qualify for something. While few people want to use this option, keep in mind, your case it is just to get you past this financial hump and it is temporary.
* Pay off credit cards. Need I say it? If you have credit card debt, stop! Stop using the credit cards and pay them off. Your cash flow is going to be so much healthier without credit cards in your life. There are entire chapters on paying off credit card debt but for now I'll say, call each CC company and ask for a pay-off program. There are amazing, God-send programs out there but you have to insist they put you on one. They'll freeze your use of the credit card but they'll also put you on a super low interest rate and great minimum monthly payment. They may even forgive some of the old debt that is so old it isn't collectible anymore.
Okay ... those suggestions are a start.
As for extra work, if you want to work from your computer, you need a secure wired line (not a wi-fi connection) that is pretty fast and a telephone headset with speaker headphones (hard to find). You also need a quiet environment - not crying babies, no interruptions, no ringing doorbells, no barking dogs, etc. With this set-up you can work as an independent contractor with Live-Ops or West (west work-at-home) as a customer service operator. Know those infommercial 800#'s on late night TV? Yeah, those calls go to these work-at-home contractors, not some big call center in Nebraska. You schedule your around your schedule. Keep in mind, the midnight shift is going to be the easiest to get. You get paid by the call, not how many hours you are on the schedule. There is a background check you'll have to pay for but there should be no other expenses (once you have the equipment). You will be expected to work a minimum number of shifts but it isn't hard. The worst part is that the work is mind-numbingly dull and you have to follow the script they give you EXACTLY without sounding robotic. Live-Ops and West (there are a few others) are not scams. They really DO pay and pay consistently. There are husband and wife operators who do nothing but work as at-home operators and do quite well. If you're really looking for big money as an at-home operator, some people work for different companies as sex-line operators. That one is hard on your marriage, though, and I don't recommend it. It is better to manage the other side of your finances.
The other work alternative you can consider is to apply to work as an over-seas contractor. This means leaving your family behind for a year or two but, as a firefighter, your skills will be desirable. Compared to how you're paid now, the pay will be spectacular. If you're looking to wipe out big debt fast, this is a great opportunity. It will be hard and dangerous work. For instance, you might be sent to the Middle East to put out oil-field fires or respond to terrorist attacks. I don't recommend this option. All I'm saying is, it is out there.
Good luck.
