Ah, the Dragon Kings... definitely one of my favorite parts of Exalted, and I never really got the chance to play with them.
The Reincarnation part was probably the most interesting aspect of them, because it wasn't along blood lines. The enemy you slew could easily be reincarnated into one of your own hatchlings, and an old friend could be reborn with a wildly different view of life. Their relationships never really settled into one form, always changing as they died and were reborn. Each life subtly flavored the relationship, one life as parent and child could follow by another as lovers, or rivals pushing each other to be better. Even your enemies, souls you hated with every fiber of your being, would never truly go away, no matter how hard you tried. They would be you enemies for thousands, even millions of years, and that made them more like a strange type of friend than anything else.
Dragon Kings got screwed by the rebellion against the Primordials, no arguing about that. They were designed to live in effective stasis. Never thriving beyond a certain limit, never truly losing anything of value. They were meant to exist in a Creation that was perfect, one that had no need to expand, or change, because it was fulfilling the purpose to was designed for.
Even if they hadn't lost the majority of their people to treachery during the war, and the Solars remained the heroes they were meant to be, I still see their culture slowly dying away. They had a set limit to their population, where the Exalted were never satisfied with the world as it was. They pushed back the borders, created new nations, new culture and creatures, pushing Creation far beyond the limits the Dragon Kings were designed to operate in. Eventually, they would be stretched too thin, spending their entire lives without meeting another Dragon King, and they would die a slow death. That, or they would have to cut themselves off, isolating themselves from the world around them, accepting that their time was long past, and that the world beyond their cities was not meant for them.
Though in all fairness, their society was already stagnant before the war. They could never really grow beyond the precepts laid down by the Primordials, their population capped by the number of souls available, making it impossible for them to truly conquer the world and crush the races more favored by the Primordials. Maybe they never learned this, because they never triumphed enough to realize this limitation. Maybe they did, but didn't care because there was never true stasis, and they could grow and change within their limits, always trying to make things better, regardless of how out of reach their ideals might have been.
In order to exist in the First Age, they needed to cast off the design the Primordials created for them. While probably not difficult to accomplish, especially given the resources of the First Age, they first had to realize the limitation for what it was. It would be nearly impossible, almost traumatic, like cutting away a part of you that had been with you for your entire life. Perhaps the Solars might have suggested this to them, but they had their own capital, and the Dragon Kings preferred the familiar cities they ruled since prehistory.
Techwise, I always assumed their technology was a lot more robust than First Age Technology, meant to be used in a world where they were not always the most powerful beings. Their devices were meant to be used, used hard, and used long. They would be simple, almost intuitive, because their reincarnated selves might not have the time or patience to unlock the deeper mysteries, especially in the face of savage opposition.
Given how central living sacrifices were to them, I could easily see each of them possessing at least one artifact that used their previous incarnation's remains as a key ingredient. Taking the strength they once held to become greater than they were before seems like a theme that would be very important to their culture. Some of them might build their homes from the bones of their past incarnations, to remind themselves what came before, and that there will be another life after this one.
Honestly, their ruins are probably mostly intact, but no one knows how to use their equipment properly, and don't recognize it for what it is. It's all just strange plants and shiny crystals to the average person. Even if they did recognize it, the Dragon Kings ruled humanity as overlords, and probably didn't want them getting ideas above their station. Some of their artifacts might have defenses or traps for the unwary explorer, and for their most prized possessions, they could be even more elaborate, meant to test that their newest incarnation is truly ready to wield the power of their past self.
I always felt that the best way to really revive their civilization would be to break the endless cycle of reincarnation. They were too bound up in their past tainted by unimaginable loss to really progress beyond it, and even if they had, they would never be able to measure up to their past accomplishments. Their numbers were too reduced to accomplish a fraction of what their previous incarnations managed, and Dragon Kings were not the type to live off the charity of others. Such weakness and dependency would be anathema to those who thrived in the most primal and savage eras of Creation.
In that spirit, I cooked up a couple of potential artifacts that could help with that revival. Sacrifice is a key component of Dragon King society, so I decided to embrace it, allowing those too weary to carry on to offer up their very existence to revitalize their people.
The Blade of Endings and Beginnings
(???) dot Artifact (either Five or N/A)
Effect: When a Dragon King willing ends his own life with this weapon, his soul is destroyed utterly. They then roll (Essence) dice, where each success represents the creation of a new Dragon King soul.
Blade of Glorious Reincarnation:
One Dot Artifact
Effect: Meant to be used in conjunction with the Blade of Endings and Beginnings. When a group of Dragon Kings willingly end their own lives with this blade, the magnitude of those sacrificed is added to the Essence roll of the user of The Blade of Endings and Beginnings.