When given the opportunity, I will always, and forever, TAKE THE SHOWER.
Here's the thing, you are sleep deprived no matter what. They tell you to sleep when your baby does. Well I have two. They don't often sleep at the same time during the day. In the very beginning, when you have a newborn, you're so exhausted, it really doesn't matter, but once sleep cycles get longer and regulate to day-time naps and night-time slumber*, when given the opportunity, TAKE THE SHOWER.
Showers are cathartic, they help you relax and organize/focus your thoughts. That's why we have so many ah-ha moments in the shower, it's a respite from the outside world. Whether it's a lengthy cleansing and exfoliation or just five minutes to wash your hair and face, you will always feel better after the fact. I always feel clear headed and energized afterwards, ready to take on the world, one to-do list at a time.
I've come to terms with the fact that I will basically be sleep deprived for the next 18 years or so, so at the very least I can be clean and smell nice. And for days where it just doesn't happen, because some days it won't, there's always dry shampoo and makeup wipes.
* For us it was about when the girls hit 10 weeks old.