I'm strapped at the moment too - my DH is working freelance and I'm living in London looking for a job. Don't feel guilty about your debts - most of us have them these days. It's not an ideal way to live, but beating yourself up will only make things worse. You don't need to give up! With a little creativity you can always find ways to save even a little money:
The car thing sounds like your biggest problem, and maybe I'm reading too much into this but I'm guessing it's your dp who's least keen to make a change here? You both have to face the fact that if you want things to improve, they simply have to change. But you can get used to new ways of doing things, and if you think 'sideways' you can come up with creative ways to save yourselves a heck of a lot of money on transport.
Can you car share - with your partner, or even with a friend (perhaps you could afford to get rid of both cars altogether if you have a friend who doesn't use their car very often).
Re car + parents, I totally sympathise. I was always taking my father to hospital appts - though I don't drive, so the hosp arranged patient transport. Why don't you ask the hospital about this, and explain your circumstances? You should speak to the receptionist of the clinics where your parents go.
Start walking more, or borrow a bike and put a shopping basket on it
Do all your food shopping in small supermarkets (not the biggies like Tesco's) and markets
It's hard not to buy clothes, but why not: (a) try 'vintage' - second hand shops; (b) create new outfits using belts and other accessories. Look at Channel 4's website for tips from the How To Look Good Naked programme. Believe me, it's possible to refresh your wardrobe without spending a penny. Maybe you could even get your sewing kit out and start customising - adding buttons/ ribbon, etc, to old outfits that you don't wear. Even ask friends for their hand-me-downs. Don't be too proud - there's no shame in not being loaded.
Energy use/ bills:
Do you have to use hot water all the time? Sometimes you could make do with cold, eg for washing hands/ face/ cleaning your home
Switch off lights you're not using
Don't switch on a heater - put on a jumper
Try to find new uses for household items, e.g. re-using food packaging for storage/ sandwiches, etc.
Cook in bulk and freeze - that way you'll save money by only reheating half your meals, not cooking from scratch every day. It will also give you a little more time every other day if you already have dinner prepared.
Food:
Always make your own lunch - including school dinners!
Cook everything you can from scratch. E.g don't buy pasta sauces - get a bowlful of cheap tomatoes from the market for £1. They'll be on their last leg so chuck them in the oven straight away with crushed garlic and olive oil. After roasting for 30 mins, scrape the whole thing into a bowl to cool - instant basic pasta sauce. Or you can do the same thing on the hob instead (roasting creates a fuller flavour).
At the same time, get bowls full of other cheap veggies for roasting - then chuck them in the oven alongside the tomatoes. Roast them with onions, liquidise them, add a bit of stock (or a cube) and you have an instant soup.
Don't buy cakes or biscuits - make them! If you google 'Guardian guide to baking' you'll find a great list of recipes from the Guardian newspaper last November. Last night I made a batch of about 40 fab cookies - peanut butter/ oat and raisin. It costs much less than buying.
Good luck!