There are several things to try. Your diapers may need to be stripped, because dtergent build-up can diminish softness. Dryer balls can help soften them up. While I can certainly understand not wanting to use conventional fabric softener, consider trying Ecover softener. It is fabulous.
With all that said, there are some fitteds, especially the bamboo/cotton diapers that will not keep that original silkiness, but the above tips should help soften them up.