I just looked up what is supposed to be happening in 2020. That Flash is being killed or dying
now are just buzz words. Adobe is just gonna stop updating and pushing Flash out. That means no increased functionality (which hasn't really changed in half a decade), and no more security updates (which doesn't matter to you who is using an offline standalone player). That
does mean it will fade away as people stop using it. But it's not like they're rendering the files or program unusable. All in all, not a big deal. It won't affect people who want to continue playing flash games, and it shouldn't affect continued work on those flash games (as far as I know).
Truth is, Flash has been outdated for years now. The sites reporting on Adobe hammering the final nail in like it matters just comes off as a slow news day.
TLDR ver: You said you're already using the standalone player. In 2020 you won't be able to update the player, but that doesn't matter. The save locations for basic SAVE/LOAD won't change either.
I would suggest backing up your saves though. Not because of this, it's just a good habit to get into.
Save Locations