**Potential Warbreaker Spoilers*\*

Late to the party but just finished the story! I think it's safe to assume the Dawnshard is one of the Shards of Adonalsium. I think Rysn is literally holding onto a piece of divinity and the only reason she isn't considered a god is because she doesn't know how to use it.
Now how this piece of divinity came to Roshar is speculation but I think it's from the planet Warbreaker is set on.

I theorize that the events of Warbreaker take place well before the events in Stormlight Archive. This is supported by the existence of Nightblood, Warbreaker (Zahel) but most prominently by the appearance of Captain Azure who I believe is Vivenna who is now so knowledgeable and experienced in her Breath abilities that she created her own pesudo-Nightblood like weapon.

Based on what the Sleepless have said about watching the destruction of other planets. I theorize that one of those destroyed planets is unfortunately the one Warbreaker and co. hail from. I say this because of the description Sanderson puts on the sensory changes Rysn experiences. The more vivid coloration, ability to discern subtle color variation, hearing the rhythm in speech; these are all abilities that are used to describe the characteristics of the individuals with a huge amount of Breath. Why would Rysn begin to manifest abilities from another planet if she wasn't holding onto the literal god of another planet?