iRule for handling traffic to multiple pools through the same VS

Define what you mean by sticky persistence?

That isn't an F5 persistence type ...

Use Cookie Persistence. Use something based on the default cookie persistence, and do not specify the cookie name (the cookie name identifies the pool - if you use a custom name, you lose the pool name information and all your pools look alike). You will have a separate cookie for each pool.

The oneconnect profile allows server-side connection reuse, which can be more efficient. However, it also detaches the server-side connection from the client-side connection after every HTTP request/respond event.

This means that a new load-balancing decision is made per request.

You may have a logical error in your irule as well - I'd ensure that any path through the HTTP_REQUEST rule assigns the correct pool for the request with a pool command.