Hi, and congratulations on the birth of your new little one. I completely understand the drain on your energy under the circumstances. I know it must be wonderful to be home after having to spend so much time with you young one in the hospital.
It does kind seem like a mixed blessing getting a second dog. On the one hand, they will hopefully play together and keep each other company, and on the other, there will be a little more work and expense involved. Finding another adult dog is also hit and miss...you might find the perfect one immediately or have to search or spend time you don't have on adoption applications, interviews, inspections, too. It is not an open and shut decision, and while finding another dog to adopt could take time you don't have now either, getting a puppy will likely be more consuming in additional ways as well. If you were thinking of getting a second dog...a playmate anyway, now may be as good as any, and in the meantime, maybe you could find some other ways to get you little furry baby exercise, etc.
I had a Yorkie-Poo when my first was born and no more demanding a baby do I believe there could ever be...my second was an angel by comparison, and both grew up just fine anyway.
My Yorkie-Poo loved fetch, so when I would feed, etc., my first son, I would throw a ball for my Yorkie-Poo. I kept a ball in my pocket and a diaper over my shoulder.
Later on, when my second was born, I used similar techniques to spend time I didn't have with my older son while nursing, etc. my younger son.
I do not know if something like that would work for you, but maybe you can think of things pertinent to your situation where you could double up on things and at least buy enough time to figure out what your final decision should be. As your little one grows, things will change in your household anyway, so perhaps the decision will become easier and more clear in the days ahead.
Good luck, and, again, congratulations!