Hi
We moved to Germany almost 9 years ago - I was in a similar position to you; practically no German, DH working from day one, had a small toddler and I was also 7 months pregnant.
We moved to a village though, so that was different, as was the toddler (22 months) rather than baby and the pregnancy.
Toddlers are a massive socialising help, and they help you learn the everyday language too, as you learn together (then they overtake you once they are 2.5 and translate for you when they are 3 and correct the spelling and grammar in your emails and notes to school for you when they are 7 
)
Are you staying long term?
My experience was of absolutely massive emotional peaks and troughs for the first 5 years, but now I am settled and despite never having managed to do more than the first 3 or 4 lessons of a German course have a "locals" type job and consider this home and would rather live here, and feel more at home here than the UK by a long way - this does have a lot to do with seeing my children flourish and be total locals with so many friends here.
Winter is hardest - we moved in early summer and my motivation was initially high, I was really quite driven to throw myself into things, but my first big, bitter, furious, hate everything, emotional "trough" came in about late January, 6 months after moving here and with a nearly 2.5 yo and a 4 month old - then things got better again for no clear reason a few months later. You will almost certainly feel better in proper spring (we are in a part of Germany that looks colder on paper than Berlin, but on our one visit to Berlin I found the technically slightly warmer on a thermometer weather bone bitingly cold - somehow the Bavarian winters are crisp cold which numbs your nose and fingers and toes but doesn't sink deep and chill your very core like the Berlin cold did! Or maybe that is fanciful :o )
I joined toddlers groups immediately - to the extent of having to look up how to ask, then ask everyone in the village playground (as little local toddlers groups aren't listed on the internet, which was my naive expectation). Once I found one people took me under their wing and it was a great way "in" - but it helped that I had a toddler who people were generally intrigued by as she was teeny tiny, bald, chubby and rosy cheeked (so looked younger than she was, like a baby) very, very active, running about and fearless, and talkative (speaking in fluent, clear and endless English sentences at 22 months and picking up German at an intimidating rate), and very sociable. In all honesty DD did a lot of my integrating for me. Again that will be easier for you in a few months when your baby is a toddler.
I never really clicked properly with any of the ex pat groups. There have been people I have got on with, but there is a feeling of everything being temporary - everyone either goes "home" or talks endlessly about going "home" or compares here negatively with "home", or goes the private international school route which becomes all encompassing and the source of social lives for the whole family at primary (and still talks about going "home" as that is often why they chose international school). However in your first few years the groups can be a lifeline. Toytown is a very bitchy place if you post asking for advice, but is a good resource to search and find groups on.
I'm with you on not wanting to go out in the evenings when you've spent all day with a baby/ babies/ babies and children - you don't have to do that to integrate, though it does work for a lot of people, it really depends. Is there any kind of facebook English speaking parents in Berlin type group? There was one for our nearest city which people did arrange pram walks and meet ups and book swaps and all sorts thorough and really did get to know one another - good if you are in a city.
Good luck - you can get through it, but you do have to put yourself out there and meet people. Don't stop going to the German groups but try a different one if the one you tried first were unwelcoming - just the same as you would in the UK!
Don't let the Oma police get you down - somebody told me they are lonely old people who can't think of any other way to strike up a conversation than to "lecture" or "correct" "young mums", and it helps to look at them with a bit of pity :o Also they only do it to young, open, smiley types I think - they don't do it to you if you have a German-resting-bitch-face (pretty sure I had the German-resting-bitch-face to end them all by the time non sleeping DC3 was a few months old, as nobody dared give me so much as a tiny hint of advice even if he was bare headed in November :o )