For one, "crying it out" REALLY helps a childs lungs. Crying it out also does not leave your child in screaming pain.. it means that when a temper tantrum, or a little discomfort happens, let the child cry. If you don't, you'll NEVER get an ounce of rest, as you are experiencing.
I, myself, chose to let my daughter cry. She's fine, perfect... very smart, she's 4 and writes her name, she's been saying her ABC's for years, she speaks to you in sentences, she knows how to change nouns naturally, she can chose to use contractions or seperate the words.
My own mother let me cry a minimum of 45 minutes to an hour every day... or every tantrum I should say. This will help teach some indepenence as well. Just make sure when you are spending time with baby, it's quality time, and they won't ever feel as if you are neglecting them.
In the meantime, while making your decision, good luck getting sleep!