I'm wondering how exactly you tried letting her cry it out. I have successfully used this method with both my boys, and with the first one it was quite easy. He didn't cry for long (at a time), although it took a few weeks before he didn't cry at all. With the second, it was different. I really wondered whether my method was not working and whether I was just being horribly mean. He was so upset, and so persistant and stubborn. He would cry for an hour! It was agony for both of us. But it took -- I kid you not -- only three days before he was not crying at all when I put him down for a nap. The key, I think, was that I stuck it out, even for as long as an hour, and that I kept going back in to reassure him that I was still there but let him know that he did have to go to sleep. I did not take him out of the crib, just went and talked to him briefly and said that I knew he didn't want to sleep but that he did have to take a nap. Then I waited longer and longer periods between my entries. Eventually, he did go to sleep, and now he's still a good napper (this was more than a year ago that I originally tried letting him cry it out).
So I would say that your daughter has learned that whenever she cries, you come and get her. If she wakes up after 20 min of napping in her crib, try waiting at least 15 min before you go to get her. See whether she quiets down. Or try setting a time of how long she will need to be in her crib. Say, an hour. Put her down for a nap and then don't go back in for an hour, even if she wakes up. Of course, if she is screaming and you think there might really be something wrong, you'll want to check on her, but just check and tell her it's still naptime -- don't pick her up.
I know a lot of people think cry-it-out is just mean, but it's really the best gift you can give yourself and your baby. Babies need to learn how to put themselves to sleep or they will have sleep issues all through their lives. It will be a few days or a few weeks of trauma, but in return you get years of peace and good sleep habits. My mother still believes her sleeping issues could be related to the fact that her mother rocked her to sleep when she was a child!
I hope you find something that works. As you say, you need naptime, too!