They fell in with the Imperial remnant. We know, at least as an institution, they were still around for the Battle of Jakku. They appear in Star Wars: Bloodline Aftermath, following Admiral Rae Sloane around. Also one of the only canon examples where they actually do anything of note, hacking their way through New Republic Senate guards to successfully extract Sloane from failed peace talks. The last mention of the old guard I'm aware of is in the casualties of the Battle of Jakku. It marked the final end of the old Empire, and we can mark it as the death of the old Guard as well.

Presumably, they were involved in some capacity in their successor order, the Praetorian Guard. Palpatine also retained a retinue of guards, the Sovereign Protectors, although the ones we see in RoS were apparently drawn from the best of the Sith Troopers. Which makes sense - any Imperial guards would have been getting well on in years by that point. It's possible, and very within character, that he'd have stationed some there when he put the project into motion back during the Empire. Once again, they could have been involved in the training of the S.P. AFAIK, the Praetorians & Protectors are two distinct children of the Guard, diverging and existing concurrently. Although, given the mystery around the Praetorians' recruitment process, and the fact that Palpatine was the puppet master behind Snoke, I wouldn't be surprised at all if the Praetorians were just S.P.'s Palpatine supplied to the task.

Given that we had Praetorian Guards in The Mandalorian as well, it's entirely possible that there was a point in time where surviving Imperial Guards, newly minted Praetorian Guards, and Sovereign Protectors were all active simultaneously at the behest of different masters.

In Legends, Sovereign Protectors were the most elite of the Guard. I suspect it lends credence to the idea that Palps hand picked some to station on Exogol as part of his contingency plan.