Is there a Women's Center in your town? They might have free financial classes or free financial advisers. I'd contact them and tell them you need help creating a family budget because you will have a life change (moving) coming up soon. If they don't provide those services, ask them who does in your area. Never be afraid to ask!
Also try your bank; you're already their customer, so ask if they have a financial adviser who will sit down with you and discuss a monthly household budget. (Be sure they don't think you want the kind of financial adviser who deals with stock portfolios and investments etc. -- well, not just yet!). I'd let a professional, who can sit down with you in person, work with you on this rather than seeking piecemeal advice here. There should be free help available if you look for it.
Meanwhile, you said you wrote down what you spent, but are you sure you're keeping every single receipt and writing down what you spend if there's no receipt? Spend a week, or a month would be better, recording EVERY single penny you spend. Don't change habits, but record the expenditures you really make and don't leave out anything, not so much as one cup of coffee or one muffin or one extra pack of cigarettes.
You will see how much you might spend on things that are not essentials: Eating out; cigarettes (are you really spending only $80 a month for both of you to smoke? I'd bet it turns out to be a more; cigarettes are pricey!); coffee from a coffee shop here and there; items at the grocery store that aren't strictly necessary (magazines, hair/beauty products beyond the basics, desserts/treats, small things for your son because they're fun or a treat). Recording every single thing you spend will show you where to trim and that is money you can put directly away each month. I know you're already recording but I mean here that you can itemize everything and judge whether each grocery item, for instance, is really worthwhile.
Meanwhile, smoking really does add up, and that's nearly $100 a month (again, I'm betting it's really more) that you can save. That's $1,200 a year in the bank for your son's education, or other real long-term needs. It's a good start if you're not saving at all right now, correct? Smoking's a luxury (or an addiction) and since you want to be around for your son, you can improve your health AND save money for him if you both quit.
And your boyfriend has to record of HIS purchases as well -- no leaving anything out! He should go too if you see an adviser. This is a family thing!
By the way, a $300 birthday party for a one-year-old who will not remember it or even realize it's his birthday is a big expense you can avoid. Reduce it to cupcakes at a park and it's almost free. He won't care. At only one, he won't even care if there are no presents; you can wrap a box with a toy he hasn't seen in six months and he'll think it's new.Then take the money you were going to spend on his party and don't put it back into your monthly budget where it'll get spent some other way; put it directly into savings and forget it's there.