So I was having the same problem when deploying Strapi to Google Cloud Run, and it turns out that Cloud Run was keeping a connection alive for every successful revision (a.k.a. every time I deployed successfully to Cloud Run). Hence I had dozens of connections alive.
I know Cloud Run keeps the connection alive to balance traffic among revisions, but in this case it was swamping my connections with inactive instances. Once I removed the revisions the problem stopped.
I do not know if App Engine works the same by keeping a connection alive for every successful deployment, but it might be worth to check if the cloud service employed keeps connections of past deployments alive as in the case of Cloud Run.