Yes you can do this and you have more than one option:
1.No Doc Loans- You can qualify even with your high debt load. There are programs available with no income requirements. These are reffered to as No Ratio or No Doc loans.
This option is the reason why there are so many foreclosures recently. Unfortunately, a lot of people got into homes that they couldn't afford. If you can live on less, and you have discipline, then this may be an option. If you don't, then I would avoid this strategy.
2. 125% Financing- You could also try qualifying for 107-125% financing. This would give you extra cash to qualify to pay off your debts, as well as buy your home.
The catch to these loans is that you will be "upside down" on your property. Which means that you will have to wait until property values are high enough to cover your loan amounts before you can sell or refinance.
3. Below Market Homes- You can find a house that you can buy for less than the market value of the home. This is becoming more common now that prices are softening.
You have to be careful though, because some properties may seem undervalued when they are really just depreciating. The way to be safe is to not consider anything a "deal" unless it is 75% or less than the market. For example, a $100,000 home that you can buy for $95,000 isn't a deal. A $100,000 home for $75,000, on the other hand, is.
You can then refinance the home that you bought and get a second mortgage to pay off your debt.
This is the best way to go but it takes a lot of work to find "deals". The best place is to start with foreclosures. I can go on for hours about foreclosures but that's not what you asked about...
If you or anyone you know needs more help with any of these topics then please email me with your contact information and I will give you a call back the same day.
I am a Mortgage Banker and Real Estate Investor