It sounds like your partner is providing a roof over your head, and that's about it. You and your son deserve much, much more than that. Still, if staying with him keeps the two of you out of the shelter, then your decision to stay put at least for the moment is probably a wise one.
That said, I think you may need to face the fact that your partner may never do the growing up he needs to do. Unfortunately you can't make someone else change; they have to turn some corner in life that pushes them to do it themselves. Now, since there aren't many more life-changing events than the birth of a child, and the birth of yours has produced no more than the lame changes you've described, I'm sorry to say that there may not be much hope for your partner stepping up to be man he ought to be. At least not any time soon.
I can tell from your message that you're a smart woman. You're articulate and kind. And right now it seems to me that you need to use all your smarts, all your gifts, all the resources in your command as creatively as you can to make things better for you and your son. My advice to you in this situation, since you can't depend on your partner, is to do anything and everything you can to empower yourself and remember that when you are empowered your son will be, too.
Use whatever, and I mean WHATEVER, services and benefits the social system provides to get yourself ahead. Use the computers at your library to research them. Call or visit your local Department of Human Services and ask them to send you all possible info on classes and other services they offer (many will let you bring your baby along). If you have a local YMCA, see if you qualify for a free membership; there are some that offer them to people on public assistance.
Don't let your situation isolate you from people who can give you help and strength. Since your partner is giving you no emotional support, find a Mom's group that can do that for you instead. Hospitals often have free support groups for new moms where babies are welcome, too -- plus you can often find bus service to get to them. Call the nearest hospital or the one where you gave birth and see what they offer. And while you're at it, find out what other classes or events you might be able to attend for free (don't forget to tell them if you're on public assistance -- there may be even be some pay classes that you can still attend free). Also think about calling local places of worship to ask what support groups they offer. You DO NOT have to join the church to attend these meetings if you don't want to. If the first group you go to feels funky, try it one time more. Still funny? Try again unless you're completely skeeved out, and then go on to find another group. There's a good chance you may even strike gold right away; if not, be persistent. You deserve to have people around you who support you, care about you, and understand what you're going through, and I promise you that those people are out there.
Consider this, too: you can change the world and your own life for the better when you put your energies into causes you care about. Volunteer organizations are usually very open to people bringing their kids along to their meetings, and they're also really great places to meet people who share your interests and concerns. So if you're into animals at all, think about joining a volunteer organization like the Humane Society or the SPCA. If litter or noise oin your neighborhood are driving you nuts, look for a community action group aimed at fixing those things. If your own experience with the difficulty of finding decent affordable housing ticks you off, see if there's a local chapter of Habitat for Humanity that you can join. Like old people? Maybe you can consider volunteering for Meals on Wheels. There's a group out there for every interest, and like I said, most volunteer groups are used to babies and kids tagging along, so you won't have to find a sitter for your son.
If you haven't already, take advantage of your local library. In addition to borrowing books, movies and CDs for you and your boy, you can use their computers for free to do research on anything I've mentioned that might interest you. The two of you can go to story hours together, and when he's a bit older your son can take part in kids' events like arts-and-crafts classes while you get a few quiet moments for yourself.
You said you'd love to find a job. Another plus of doing all (or even just one or two) of these things is that research has shown that the best way to find the job that works for you is to meet all the people you can. And the best people to meet are the ones who know lots of other people and who are involved in lots of public organizations and activities.
Whew. I know that's a lot to swallow. If these ideas seem a bit intimidating to you at the moment, they'll be easier if you take them a step at a time. If you're shy, remember that you need to be a bit bold now for you and your son. Your family needs a leader, and considering your partner's unwillingness to do more than the bare minimum, that leader is probably going to have to be you. So don't waste energy fighting with him. It's not good for you or your baby, and it sounds like it may not do much good anyway.You have an unquestionable right to ask and expect your child's father to do better, but since he's so resistant to putting out any more effort than he already is, it may be better just to start turning the world yourself.
I hope you'll get out there and grab all that's yours. I hope you'll get in mind what you need and ask for it. And ask for it again. And keep asking whoever and wherever you can think of asking until you find the path that takes you where you want to go. I really, really believe that you can do this. And who knows? Maybe you'll show your family how wrong they were when they didn't believe in you. Maybe you'll set an example for that man of yours. Maybe you'll be the inspiration that gets him off his behind and someday makes him a better role model for his son. Or maybe you'll meet someone who deserves and will treat you better. No matter what, *you* will be the light that leads your boy to a good and happy life. I wish you all the luck in the world.