to clarify about fleece/microfleece...fleece comes in several weights. Microfleece is thin and thus wicks through...thicker fleeces are absorbant but stay wet (in the case of hemp/bamboo) or else are a barrier/cover (if polyester). Fleeces are really soft, which is nice, but IME they will get a little pilly with time...
If you want a stay-dry, try microfleece or suedecloth. If you want a feels-mostly-dry-but-technically-isn't then consider velour! The fuzz is tall enough to basically feel dry, even when it isn't.