I just got home and caught up with these posts. Whew! You have been given some good advice here. Please take the time to reread and ponder what has been said. Glad you are getting your name on the account. You need to take a serious look at the money situation. I added up what you said your bills were and based on the $3400 coming in and the bills you listed, there is $1071 that is unaccounted for. Granted that there are some other bills you have forgotten, but that is a lot of money to be unaccounted--where is it going? There should be enough for you to get clothes and to buy the baby supplies and even buy a pizza or two.
Please, please, please look at other birth control options! You do not need to add any more babies into this mix right now. I have read what other posters have said about getting your DH to help, but I think it really boils down to the fact that he told you he didn't want children! I don't mean to be harsh when I say that, but that is what you said early on and throughout the posts. It didn't magically change for him when they came and it may never change. He may have been happy with just the two of you. Maybe he was okay with the one baby, but now there are two. Yes, he got you pregnant but he thought you were on birth control that worked. Look into getting your tubes tied or using mutliple forms of birth control from now on.
I think you need to get rid of the dog as many have suggested, even though you don't want to. I know it is hard to get rid of a beloved pet (we had to do it in one of our moves), but sometimes it has to be done. You have 2 little ones to look after and if you would ever have to leave, it will be hard enough with 2 little ones without adding a dog in the mix. Find a good home for the dog-let the poster who offered help you or go to the bassett rescue.
Look at your finances and get rid of everything you don't need--cable tv, go to a cheaper internet, etc. You need to build up a savings account for "rainy weather" or you will really be in a bind if anything bad happens--dh loses job, major medical issues, etc. The perfect time to make these changes will be when you move to the new apartment. Hope you like the one with the w/d as that would be so nice for you to have in your house rather than taking 2 little ones and all the laundry to the laundromat. Also, if you have a w/d, you should do dh shirts and give yourself the $20 a week as your "fee". I know you will have to little ones, but you could do this when they nap or after they are in bed at night. Then you would have a stash to use as needed and to pamper yourself once in a while.
Again, good luck.