Hmmm. I think people are being rather harsh. I can understand why but it's not helpful.
OP, it's clear to me that you don't want to live off benefits, and you're trying to do things differently from your mother, but you're feeling a bit trapped atm. I have some suggestions that I hope will help you to feel less trapped:
Firstly, your PND and anxiety is difficult as it is but it's clearly affecting your ability to find work as well:
"I feel like I am in a deep dark hole. I get so scared about what the future holds and if I am ever going to find a full time job I like, but it's been so long since I had a full time job that I'm scared of interviews, meeting people etc."
I think you could do with more support on this. ADs may help but they won't fix it. Could you ask your GP about CBT or other counselling services that might be available? I do think that would really help you to feel better about yourself and your ability to find work.
Secondly, the practical and financial situation. You've mentioned JSA, but are you or your partner receiving any other benefits, such as Housing Benefit, Child Benefit, Council Tax discounts, etc? I advise you to find out whether you are entitled to any other benefits while you find your feet. Also find out what will change if and when you and your partner get more work - you are likely to be eligible for Tax Credits, for example. When you do the sums, you will probably find that you'll be financially better off working. Depending on the job, you might be able to get childcare vouchers from an employer. And have you looked into all the childcare options?
Thirdly, you would probably benefit from some decent careers advice. The job centre are rubbish when it comes to that kind of thing. Why not look up the National Careers Service and see if there's an office near you?
Lastly, and perhaps most importantly, you and your partner need to be working as a team. If he is only studying for 12 hours a week, he can do childcare and/or get a part time job at other times. You are both parents and both need to take responsibility for the childcare and the finances. I suggest you sort those out before you consider having a second baby.
As for your partner saying you need to lose weight and tone up... That's completely out of order, especially given that you're struggling with PND and anxiety and you're the only one doing some work to support the family. He needs to show you some appreciation, respect and support.