The basic foundation of CBT is the assumption that 'Your behaviour/actions influence and shape the way you feel'. The way you behave can set your mood, in the simplest example: a brisk walk in fresh air, a tidied room or any form of self care might uplift you.
Do you like that idea? Are you currently able to motivate yourself to do things, starting with very small changes?
If that speaks to you, then CBT can guide you onto a path of positive behaviours that will then influence the way you feel, making you gradually better and better.
CBT can support you changing your behaviour and state in the here and now starting a positive spiral of getting better and better - without examining the root cause of WHY you feel sad.
But you could reflect and journal that out yourself.
OR with the counselling you'll look at the root causes and help you analyse those (and hopefully that will motivate you to coach yourself into more positive behaviours).
CBT can be one of the more effective and fast therapies at getting patients out of a dark hole, but it all depends on what philosophy you prefer.
I assume the NHS has done all the due diligence regarding possible postnatal depression? As that might require a more medical treatment (and I'm not qualified at all to comment on this but I know that it would be good if your doctor checks this thoroughly and keeps an eye on you)
OP, sending you lots of love, having little children and depression is so challenging and I'm so glad you want to do something to feel better
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Do let us know what you decide.