I'm working with CartoDB and hoping to automatically generate labels that can then be moved around the map. I'm able to generate the labels, but it won't let me edit their placement.
You would need to open the CartoCSS wizard of the CartoDB sidebar menu to edit the position of the labels. In order to change the labels placement, you could use the CartoCSS properties text-dx and text-dy. Here you can find a lot of text properties to use with CartoCSS in CartoDB.
You can also find a lot of information of placing labels with CartoCSS here