I keep looking at these myself (I like the plaid version) but the washing instructions have made me hesitate too. I'm a mom of three including a 2 year-old and I enjoy and wear my clothes that are easily washable more than those that require special care, including even a special gentle cycle. (I do hang dry many items and find that isn't inconvenient.)
For a new mom, I would definitely err on the side of easy care. Everything gets messy from everything imaginable. I found I couldn't wear anything more than once and sometimes had to change my own outfit during the day! Babies make so much laundry!
Here are some cuddly fleecy things that would work well for a new mom, nursing or not. You might find that more gear-oriented shops/departments might have tops that wash well.
Also, I personally found nursing bra tank-tops perfect for breastfeeding and would wear them under just about everything.
http://www.cuddlduds.com/women.....-wrap.html
http://sahalie.blair.com/p/wom...../sc/1923/9
http://oldnavy.gap.com/browse/.....=119918162
http://www.gap.com/browse/prod.....=140299012
http://athleta.gap.com/browse/.....=918829002