Eliza, everyone has provided really brilliant advice thus far, particularly Gin - we are listening to you 
. Being told to go and get help can sound very overwhelming and scary - believe me, I know.
Let me tell you what a fantastic Mum you've been already:
- You appear to be more or less bringing your children, one of which is very young, up by yourself
- You've done all of this whilst suffering from anxiety, and your own pre-existing health issue (diabetes)
- You love your children dearly, which is why you are worrying so much
- You've found you may be struggling, but you've kept on chugging away, and have now sought help for how you are feeling
Anxiety is like a parasite that coasts through your veins, polluting all rational thought. It almost feels like your suffocating, doesn't it? It's a bloody nightmare at the best of times - I struggled for a while to get a hold of mine and I was a single woman with no children! I can't begin to imagine how difficult it must be with one child and a newborn in tow! So, for this, please give yourself some credit - you probably won't, and I understand that, because I'd never give myself credit either. But that credit is yours, OK? 
It seems as if the GP appointment has helped you address some of your worry, but it hasn't really helped the fear you have towards your DD's health? Does that sound right? This is the thing with anxiety - we can take all the medication, we can seek all the advice, we can talk and talk and talk until we are blue in the face and exhausted, but ultimately, unless the anxiety is addressed, all the fears and worries and thoughts won't go away.
This is why, my love, your concerns are based on anxiety, not illness, not chicken pox, not vomiting bug - it's all about the anxiety - it could be any illness, any life event, any cataclysmic event, but the anxiety will ALWAYS win out. Because that, Eliza, is the (anxiety) illness - it's hiding there, trying to tell you to worry about everything else, convincing you that chicken pox, vomiting, DD being premature is all YOUR fault and that you're both going to die. The only thing that you need to deal with right now, for yourself and your beautiful children, is the anxiety.
Anxiety is like a really shitty ex boyfriend/girlfriend/friend who blames EVERYONE else for something they've done. It's bloody annoying, and after a while you cut them loose and move on. You can do the same - antidepressants (I don't give a fuck about what your DH says, if a GP has recommended them, take them), being open and honest with your GP/Perinatal Nurse like you've been on here, seeking help for Mental Health, either via your GP (therapy) or through counselling, and being KIND TO YOURSELF.
If you trap yourself in your house, DS and DD included, you are allowing the anxiety to affect you AND your children. Children need to be out and about, seeing the world, being exposed to life. It feels like keeping them indoors protects them from the ills of the world, but all it's doing is transferring your anxiety to them, and that, Eliza, will really affect them, both through childhood and into adulthood. Trapping yourself in isn't addressing anxiety, it's giving into it and allowing it to control your life. Believe me, anxiety (like the shitty ex boyf/girlf) is NOT worth it 
.