It is harder to find a job with a felony conviction on your record, but not impossible.
If you are being treated that badly by a state agency that is supposed to help you find a job, then look elsewhere.
A friend of mine had a similar problem and struggled to find a job for a long time. His solution? He signed up with a few temp agencies. That got him work, it was only a few weeks in one place or a few days in another, but it was income coming in and it showed people that although he made a mistake in the past, he was working hard to rebuild his life. In the times that there were no job contracts for him he'd make money mowing people's lawns, painting houses, and doing odd jobs here and there.
Long story short: He worked for a few weeks at an office on a temp contract and the people there were so impressed with his work ethic that they offered him a permanent job, knowing that he had that felony conviction. He's been there for three years now and going to college at night, but most importantly he is able to support himself and isn't a mooch.
A lot of times, looking for a job and getting it are more of a job than the normal 9 to 5 grind and sometimes it's a lot harder for some than for others, but you've got to do what you've got to do. In the end, it does pay off.