I can honestly, absolutely 100% understand your anxiety- I went through a very bad period of post-natal depression 11 years ago, with very similar symptoms. I was fearful that me or my baby would die; developed OCD regarding cleaning and the counting & protection of sharp objects, and lost chunks of hair due to pulling it out. Unfortunately, it was the start of a good few years of depression. I believe it escalated because I didn't get the right help to begin with. I eventually had CBT (Cognitive behavioural therapy) with a psychologist and swear by it! It saved my life, to a certain extent. However, since then I've also qualified as a therapist & now teach psychology, so my next advice/comments are from both sides of the fence.
The anti-depressants/ anti-anxiety medication should only be short term. Depending on what they are, they will most likely calm down your extreme symptoms and slow down certain other thoughts. Don't be afraid to take them, but be aware that meds for anxiety and depression should only really be a short term measure. If symptoms persist after around 6 months- 1 year- then therapy is definitely advised. The medical model of treatment (i.e, medication) is the most common used by doctors- hence, your pills. The reason being is that the medical profession still very much adhere to the belief that depression (particularly post-partum) is due to fluctuating levels of serotonin and dopamine in the brain. With post-natal depression, these levels of brain chemicals are affected by changing oestrogen and progesteron levels after giving birth. As post-natal depression is thought to have a chemical cause; medication is regarded as the quickest way of treating it.
However- there are explanations and treatments that don't adhere to the medical model and that don't include medication. Cognitive Behavioural Therapy, REBT (rational emotive behavioural therapy) and relaxation therapies. What kind of treatment that works best would depend on other factors, such as whether you've experienced anxiety/depression prior to having your children and other symptoms related to your anxiety and depression.
Perhaps having a chat with your GP again? Sometimes, you have to ask to get therapy- not all doctors will offer it when 1st treating a patient with anxiety/depression symptoms.