Can session persistence time impact load balancing

What kind of application? HTTP Traffic?

If you disable a node on the BIG-IP, persistent session can still connect to this one. When set to Disabled, a node or pool member continues to process persistent and active connections. It can accept new connections only if the connections belong to an existing persistence session.

If the service goes down, e.g. the pool member becomes red because the check fails, the persistence will not create your issue. It will get lb to another available server no matter what kind of persistence you use.

Your issue sounds more like this one. K15095: Overview of the Action On Service Down feature so maybe give reject a try.