I pretty well agree with everyone else: you just need to keep whittling away your budget as much as possible. At this stage you are on emergency debt payoff mode, so you need to be brutal. If you can for example just save $10 a day, every day for a year , thats $3650 and the credit card gone.
Debt wise I would go for the Credit card first (assuming its got a higher rate than the car loan). Then the car loan/emergency fund.
Budget wise:
Car: Is there anything you can do cut this down? Could one or other be sold and a lower mileage car purchased? Can hubby commute/bike? Or work from home even 1 or 2 days a week. Your petrol costs are huge even in Australian terms. Could you both learn/practice hypermiling techniques?, really think about all your trips and try combining/reducing mileage.
Phones/Internet: slash this in June...you've only got a month to go and you can be busy fixing all the other stuff til then. At $240/month you could really make big savings here.
Electricity: Seems high especially for a more southern state. Check out
http://michaelbluejay.com/electricity/ which I think it the best overall guide on how to cut down your electricity usage. For a family of 5 you are aiming for <25kwh/day at a gentle level of frugalism. (experts at this get under 10kwh/day). Small changes are worthwhile: everything adds up.
Kids allowance? Do kids under 5 need an allowance? Not when you are in emergency mode
Food/diapers isn't too bad, but Again this is an emergency so scrutinize everything, maybe you could knock a bit off here. Cloth diapers yes, but line dry, no dryer.
Xmas fund: agree use to pay off CC. Also what about trying simple handmade gifts as a gesture? There is a thread on hear somewhere about home made gift ideas. I know 3 boys under 5, you have your hands full, so something simple!
Chiropractor? might be essential..can;t comment
Personal:$20/week each sanity money , not too bad, but this is an emergency! If you are not using yours why not deliberately pay it on the CC so you can see it being useful, rather than let it evaporate into the other categories. As for hubby, well you don't get coffee and ciggies when your hair is on fire or you are being stung by a cloud of killer bees...but if he really is unmovable on this its not a disaster...just another $1000 a year you don't need to spend.
I know from your other post you are feeling a bit frustrated...those unexpected bills happen to us all just as we think we are making progress.
Overall though, your debt is not huge, and we all see areas of wriggle room/hope in your budget. Try going all out for every cent of saving until those debts are cleared....