Definitely start socializing now. We go through this with each new senior who joins us. We have grans from 6 months old to 17 years old. They are told in no uncertain terms, when there is a new dog in the house, that they are not to approach the dog and we watch very carefully. Bandit had never been around a baby and when a new gran was introduced, he was not certain at all. When the baby started crawling, he actually rumbled a little. He was allowed to pick his spot and the baby was not allowed to approach. He was gently corrected for rumbling. As soon as the baby stood up, he seemed to click with him, hey, it's a human.
Potion, Melody and Mardi have been furry mommas with entire time. When introducing, they are allowed to sniff while we supervised. They have had no problems at all.
We were unconcerned about Archie. He supposedly had been around small children but with all the other lies we were told, we were not sure. I actually brought home a blanket from the hospital when Little J was born and put it on the end of the couch. It stayed there for a good week. He was allowed to greet by sniffing. It's gotten now that we just have to be sure they don't step on the baby if he's on the floor.
I would get together with a trainer and set up some situations for Jade to help her gain confidence. She sounds unsure and needs to learn that babies are good.