Is there a reason you were trying to wake him up? Were you concerned about him and trying to check he was ok? I don't see why you need to be embarrassed - if they called the ambulance they were clearly worried and so it's not like you were over reacting or anything?
If it's that you were trying to wake him so he could be present for the class I really wouldn't bother. He isn't going to care one bit about it at that age.
Six weeks is when (for some babies) you start getting smiles but not consistently. Also it's still the stage where it really is still potentially wind and you gradually start getting more sure that no that was definitely a smile. The bright eyed, lots of looking and smiling and not sleeping as much babies sounds like they were older.
But also don't compare yourself or your baby to others. Mine had CMPA and reflux and I used to see people do things like lie their babies flat on the floor or just take them out for coffee with friends without them screaming for literal hours and they'd just go about their normal lives and I'd feel so jealous and assumed I was doing something wrong. But he was just a difficult newborn. Once he got to about 8/9 months he was easier than most others IMO.
Go to these things for you and choose the ones you like. Let your baby sleep as they like.
I actually think the cheap/free stay and plays that you just choose whether to attend each week, usually in a church hall or whatever are better than the sensory/yoga/massage branded ones where you pay for a term upfront. And even better yet have been the free children's centre ones - they have been excellent - but they are only an hour which is a shame.
At this age you just need to keep him alive every day and let him enjoy hearing your voice and feeling your body against him. Outside of that, do the activities you like best.
As far as interacting with him, I wouldn't put too much pressure on yourself with this. Mine had the audacity not to coo at the exact minute the milestones prescribed he would and the the HV told us to talk to him more. We had already been talking and singing non stop and at this point felt we had to become full time singing and dancing children's entertainers to this little potato.
I get it. As far as she knew, I might be one of those people who just played on my phone ignoring him all day or left him crying to "toughen him up". But I wasn't and if anything I was probably stimulating him too much. I feel like there's so much pressure and such high standards and it really gets hard. You're evidently doing great with him.