I feel like this is either a reference to the Recreance, or to the Skybreaker's specifically, who didn't partake in the Recreance, and have since turned to Odium's side.
I have to concur that this one is an everlasting weeping, or even something similar, but not just rain persistently pouring down. Or I guess it could be that something major happens during the weeping. Do we know what month the weeping happens(Is Palah the last month of the Rosharan year? Seems like a coincidoink to me.) The timing just adds up I think.
I propose that this is from the singer's perspective. Somehow those who serve with Dalinar all get sent to Braize. (Maybe some trickery will occur with the contract.) Or this could be a renewed oathpact(see below).
Humans burning land behind them as they retreat, breaking the contract and subjecting them all to Braize.
A new oathpact. Heck, it could be Dalinar and 9 other people who become Odium's oathpact and are forced to keep the human's of Roshar penned up on Braize. Or, and this is kind of out there, it's some of those Rosharan humans who managed to forge a new oathpact while on Braize, and this is a reference to the events of the back 5 where they free themselves.
I think this is the Recreance again, or a new recreance when the humans get kicked to Braize. (I know, I'm really leaning into that theory, but it really feels like these death rattles are implying it.
Obviously the orbits of Roshar and Ashyn align when the endless weeping begins, such that Roshar is "ahead" of Ashyn , but soon falls "behind" (As Ashyn has a shorter orbit that Roshar). Maybe someone is still on Ashyn, even after the exodus (is it that much of a stretch see spoiler below), and they end up being important to freeing the humans stuck on Braize.
AA (Sorta SP4 too) spoilers
Nightwatcher ? At the end of SA10, Odium has been splintered too, or escaped Roshar, leaving Cultivation as the only shard present.
Spoilers for certain snippets of unreleased books not to be named here(for fear of accidently giving away important points)
Obviously these are pretty wild guesses, and honestly, it's a really dark outlook, so please poke holes in it. Just be nice, I haven't done real theorycrafting in a while.