It might be worth speaking to your GP again. There are loads of different anti-depressants so it’s a bit of trial and error as to what work, citalopram made me really nauseous but sertraline was fine - others find the opposite is true.
If you don’t want to take medication you could go back to counselling/therapy. Some folk find CBT helpful (and there are exercises/worksheets you can download which are sometimes helpful) some find a more relational therapy more helpful. It’s often not a “one and done” thing with therapy and different types can help at different times.
When I felt like you do just now someone suggested I go in and look at my kids while they’re sleeping. Not only do you see how small and vulnerable they are, but you also get reminded of how wonderful they are (in peace, when there are no demands being made on you). It might also be worth thinking about a gratitude journal - not because it’s a wonderful magic wand, but it orientates you to looking for the good, so you notice more of the positive things around you.
You also sound exhausted - with good reason. Is there any way to relieve some of the burden on you. Maybe some childcare, or coffee with a friend who might help corral the kids for a while. Make things as easy as possible for yourself, which may mean lowering your standards a bit (even if you think they’re already low, if you’re not coping they’re too high). I had an “everyone fed, nobody dead” mantra for a while. The house looked chaotic but everyone was fed and clothed, everything else could wait until I had some help.
Its a horrible time, please don’t beat yourself up, you’re not failing you’re trying your best.