Congrats. I'm about half a year into it. A little bit older than you, but not by much.
I've been handling the broken sleep pretty good (usually 5 or 6 hours, then 2 more hours) but probably mainly as a result of my mileage dropping from the 70s to 45-55 because of the reduced time available for running in the morning since I am in charge of daycare drop off. The time I've had to run each day has shrunk from about 90+ minutes to usually around 30-40 minutes but sometimes up to 70.
The time crunch and slight unpredictability of the day-to-day running window has caused me to prioritize flexibility in training. What I mean by flexibility is that my goal is to be ready to workout any day that is not the day after a workout, since the 60-70 minute days could come at any time. That has caused my workouts to go from more "80-90% efforts" that might leave me fairly tired for a day or two, to more "60-70% efforts" where I almost always feel ready to go again by the 2nd day after the workout. I don't think I have been sore or tight waking up the morning after a workout since baby except for after the first time I did hill sprints. The time crunch also basically forces me to make all my non-workout runs short and easy, aiding in recovery. Is this the most efficient training method? No. Does the mindset make it easy to enjoy running and maintain some semblance of fitness while also having the time and energy to enjoy raising a child? So far, yes.
I don't think I've gotten "more fit" (I haven't raced yet), but I feel like I have done a really good job of at least maintaining a baseline fitness where if I hopped in a race tomorrow, I might not be in it to win it, but I could put forth a good effort and not totally embarrass myself. And if the opportunity presented itself to get in a month or two of focused training, I would be well prepared to make the most of it.
Only real setback has been 2 cases of Daycare Disease, made me feel like doodoo for a couple days each time. Nyquil is like a plunger: you want to have it before you need it. This applies whether or not you have a baby.
When there's a crazy night, I don't feel bad about taking a day off. In the grand scheme of things, It's all good.
If the stars ever align and you've got the green light to go out for a nice long run on the weekend - take it and savor it!
Good luck!