The tear away can be taken completely off except for what is under the stitching. When I do flour sack dish towels, I set up my desktop sewing lamp and put my magnifier around my neck and get out my pointy tweezers and pick the stabilizer off until it is gone. I want it all off before giving them as gifts as I do not wash them first.
As a side note about Dreft--not every baby's skin can tolerate this. My kids and grandchildren all broke out terribly from it, as did a couple of nieces. We found out if we washed the babies clothing in the same mild detergent we used, their skin conditions cleared up considerably.

Thanks Nana. I was just figuring that most new Mom's are advised to use Dreft so I figured she would have that rather than me washing them in my Tide! I ordered a product on Amazon which should do the trick.