I second Matt's recommendation. Lease-options and lease-purchases (especially lease-options, in today's market) are ideal for what you're trying to do. It lets you get into a home, lock in a price, build equity (through monthly option credits), and restore your credit. And you can get into a lease-option, often, with no more money than it would cost to rent--the equivalent of several month's rent. Make sure the money is credited to the purchase price, though, in the form of option credits. (I lease-optioned a property while the market was still hot--May 2006--for no money down. It's really not difficult.)
There are a bunch of other ways to buy a property without getting a new mortgage. Some wouldn't be right for you, but some might be. You might want to research (Google) "subject to," "contract for deed," "equity sharing," and "owner financing" to start. Post here if you want further details on any of those.
You definitely can do it. Good luck.