Dear Sarah,
It´s nice to hear good stories from people moving to Spain, I was in the same position not long ago, so I know everything can be a little bit too much. Congratulations on the baby and thank you for the kind words with regards to SOS. I am the official interpreter of the company, so we are very happy when we hear about satisfied members. I was looking at your E106 situation and found the following info on the NHS webpage. Probably the info will be useful. http://www.nhs.uk/NHSEngland/Healthcareabroad/countryguide/Pages/healthcareinSpain.aspx
On the page click on Living in Spain.
I know from my experience that unless you register with the Social Security (that is paying taxes) you don´t need to register with the local clinic, and if you need to go in you show the European Health Insurance Card or the E106 form. Also, the 112 services is free for everybody. I have registered autónomo (self-employed), I pay a monthly fee to Social Security, is 179 euros, because I am under 35 years, and I benefit from national health cover and pension. So, you should take the papers to the Estepona Social Security first and hopefully they won´t need anything else.
Follow the steps in the link, try and find out your exact situation at the Estepona medical center and if now drop me an e mail and I will see if there is anything else we can help: valentina@sosmedicos24hs.com. I belive it is very important you ask them also what will the baby´s situation be, what do you need to show to get him registered with the clinic.
As you know our 24 hours doctors are available and happy to attend, but you should clear up the situation with the local hospital.
Hope I have clarified a bit the situation and as I said, any questions, don´t hesitate to contacts us!
Kind Regards
Valentina