If you decide to move off of RDS but stay on Postgres, you can run your own Postgres but now lose all the benefits of a managed service. You could move off of AWS (eg to Azure), but moving to a different cloud is a huge lift. That, btw, would also be required if you want to try something like Spanner (move to GCP). Moving off of Postgres to another database is also risky. The migration will obviously take some effort, but you're also probably talking about lots of code changes, schema changes, etc, as well as unknown operational risks that you'll discover on the new system. This applies if you're talking about things like Cockroach or even moving to MySQL.
That said, rolling your own sharding is a MASSIVE undertaking. Limitless looks promising, since it takes care of a lot of the issues that Figma ended up spending time on (basically, you shouldn't need something like Figma's DBProxy, as shard routing, shard splitting, etc will be taken care of). It's still in preview though, and like the article mentioned, the costs may be high.
Overall, no easy decisions on this one, unfortunately.