Are there fonts that will work for "writing stlye" with Cricut explore? If so, how do you tell which ones they are? Can you filter the fonts by something to get the ones that will work?

Cricut uses CAM software which is something totally different from what is needed for print and/or on screen (ttf, otf, fon). What you need is software that can convert ttf/otf into a CAM format Cricut can work with.
I remember reading that some cutter/plotter machines and some embroidery machines can do this but I have no idea whether Cricut can too.

By definition, TTF and OTF fonts can not be "open path" like Cricut writing fonts. However, if you choose ultra thin fonts, you can mimic the look of the open path fonts because the outlines collapse on themselves depending on the size you are writing and the width of the pen tip. I have links to lots of fonts that work well for writing/drawing in a diecutter here http://cleversomeday.wordpress.com/ttfs-for-diecutters#penfonts

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Try https://cleversomeday.wordpress.com/ttfs-for-diecutting#penfonts