OP, you do sound a little depressed. Please go and see your GP and Midwife. You have nothing to lose, you may be struggling with antenatal depression, which is often nonpt picked up as feeling tired then low when pregnant are normal right. Personally from what you describe I think it's a bit more. I have had PND and a couple of mild bouts since and the odd burst of anxiety so I can relate to how you feel.
Try to get out of the house every day, even if just to shop or park - weather picking up!
Just do basic, feed, clothes bath and bed. Choose easy options.
Some perspective.
I also think it easy to feel likel this when pregnant, you tend to focus on the whole parent thing and it can be quite a reflective time/take stock etc. BUT when you are pregnant with young children your options are still there just harder to access or arguably limited. So it can lead to frustration as you want to make changes but can't, make plans instead.
I remember reading on MN if your kids are still alive at the end of the day and in bed then you've done your job as a parent. I have also read about the 'good enough' principle, setting your own achievable standards.
For now:
Ok so they will have to share a room most people on there third or more child have this - can you alter the room to make it into big boy/girl den? I know you don't have any spare money but is there something you could sell to buy new curtains or funky storage - make the best of what you've got might make you feel better.
Re your housing - in the short term the influences you speak of are able to be kept at bay so to speak, this gives you time to plan a move, as someone up hard said, I would throw everything into this. BUT FWIW you are and will be a major influence in your child's life.
Have you considered studying part time, I do this with OU and it keeps me sane whilst retraining ready for when my 1 yr old goes to school. There are many levels to get in at, have a look, with a longer term view for your own job prospects.
I think the fact you are thinking these things through and posting for views show what a thoughtful and responsible parent you are.