It's an Australian Terrier, not an Australian Shepherd.

(Though agreed, both are hearty, working breeds.)
I had to look them up because I didn't know much about them, other than in being a terrier, I knew they were bred to go after vermin.
And I must say, I'm impressed. What a neat little dog!
http://www.dogbreedinfo.com/australianterrier.htm
This story does lend credence to why problems can develop, but I don't think it's a futile situation and I wouldn't rush out and rehome the dog just yet. This dog has the potential to become a great family pet and protector if it's handled right, now.
OP, you and your DH have GOT to incorporate your dog into the household even though I realize it's challenging with a new baby. There should be Baby Time, and Dog Time, and even Baby and Dog Time - not just Baby Time. We brought home two babies with two active German Shepherds in the household and we worked hard to see that everyone stayed healthy and happy. Not only can it be done, it can turn out really great when your child grows up having a wonderful, well trained companion.
There will be special considerations with dog and baby because of the breed (not unlike many other breeds who have their own considerations). Terrier instinct is to go after little things. So your job now will be to train both your dog and your baby to respect eachother and live happily and safely together.
Here's how I would handle it, as an example. Tend to baby's needs. Tend to dog's needs (exercising and toileting; this will work best to start if dog is already tuckered out - always remember A Tired Dog Is A Happy Dog). Tend to your own needs. Then put baby in swing or chair or whatever and play with dog in front of baby. Baby will probably enjoy watching dog chase ball or toy. Let baby participate by throwing ball or such. Do this at least twice a day. Keep it relaxed and happy. Everyone will come to look forward to this time eventually. It may sound like a lot of work but it's actually more relaxed than the chaos of an undirected dog running underfoot. Build on these games as both grow up, always keeping things under your control. There is no limit to what your dog can learn to do if given the structure, time and attention.
Work on your obedience at this time by having dog obey commands during play time (which is good to teach during distractions). Never allow baby >> toddler >> child to disrespect dog in any way (and obviously vice versa). An adult needs to be there with each on each and every occasion they're together. (Later, after mutual respect and familiarity are learned, this can be relaxed somewhat, but there should always be good supervision when dogs and children are involved.)
I favor this approach over a doggie day care situation for a few reasons. A) financial. B) Not all dogs at daycare are well trained or supervised and your dog will both miss out on that and can learn bad habits. C) I was talking to a friend at work this week in this same situation as you (with a big, active Lab who previously lead the Life of Riley but now, with new baby, is bored and underfoot) and she made the appt at a doggie daycare but thankfully read some reviews first and found a lot of complaints about this place including one person coming to pick up their dog and finding two teenagers watching the place and a shock collar on their dog

and another reporting seeing the teens hitting dogs with shovels. Not a situation I would ever want my dog (or any other) to be in.
I know life gets stressful when babies come along, but make this time for your family (and yes, your family includes your dog) and it will pay off in the long run.