Don't you just love MIL and mum's comments! If only they knew... You sound like you are doing a really great job and going through a rough patch. If DS is screaming when out he is prob overtired. FWIW my DD (now 17 months) won't sleep at anyone else's house, in the car or in the pram when out. Some babies are just like that and she has been all along. Get out when you can after DS has had a sleep and feed and don't stay out too long and I would avoid really stimulating places with lots of people ie shopping centres!
Another great idea is a sling. I got an Ergo carrier when DD was about 4 or 5 months and I still use it every day. She loves it and no matter how tired she is she never cries in it and I can take her every where in it. Babies feel really safe and secure snuggled up next to mum's body.
Your DS probably sleep 3 times a day??? I can't remember the sleep at that age but I do know that 45 minute naps are very common at around 4 months onwards even though bubs needs more and wakes up cranky. Perhaps just stay home for a few days till you find a rhythm that works for you both and that you can manage outing with.