This is significantly affecting us as well, and as of now (Dec 1, 2022), still no word on Data API. Also, per someone else's comment, Postgres v10 on Serverless v1 IS being force upgraded to Postgres v11 in Jan/Feb 2023. We've received multiple notices of it. Unfortunately for us, this comes with an update to Postgis v3.1, which is a dramatic performance decrease in our usage (vs. Postgres v10 + PostGIS 2.4 that we have now). We've done tests, and basically it destroys our ability to use it (went from sub-second query times, to some queries taking nearly a minute!).
So, our only path at this point is to move to Serverless v2, which allows Postgres v13 or v14 (we'll go straight to v14.5), which does NOT have the performance problems we saw with v11. But, we were fully using the Data API, so not only do we have to deal with that, but it means putting all our Lambdas back into a VPC, incurring the NAT Gateway cost (minor in the grand scheme for us, but could play for others), and of course just the higher complexity of all that.
I find this very disappointing on AWS' part - that they want everyone to move to Serverless v2, but they didn't create feature parity (Data API). I welcome the ability to move to a much newer version of Postgres, but am very bummed about the lack of Data API and the VPC requirement, etc.