Does Lowering Priority help in draining connections for maintenance?

Depending on the setup of the priority group activation, lowing the priority of a pool member could cause the LTM to add members from another priority group. Example, say your pool is setup with the config below. If you move node3 from priority-group 10 to priority-group 5, because there are now only 2 active members in the priority-group 10 the LTM will include all members from priority-group 5.

If you are looking to move the traffic off a pool member for maintained, the fastest way is to “Force Office” the pool member, which is only accessible from the pool member page. "Local Traffic...Pools......Members Tab..." Select Force Offline...Click Update. Forcing offline the pool member allows only active connections to that pool member. If you “Disable” a pool member from the pools member tab, this allows persistence and active connection to continue to that disabled pool member. "Local Traffic......Members" Select Pool members you want to disable...Click Update.

Example config:

ltm pool priorty-group {
    members {
        node1:http {
            address 10.0.0.1
            priority-group 10
            session monitor-enabled
            state down
        }
        node2:http {
            address 10.0.0.2
            priority-group 10
            session monitor-enabled
            state down
        }
        node3:http {
            address 10.0.0.3
            priority-group 10
            session monitor-enabled
            state down
        }
        node4:http {
            address 10.0.0.4
            priority-group 5
            session monitor-enabled
            state down
        }
        node5:http {
            address 10.0.0.5
            priority-group 5
            session monitor-enabled
            state down
        }
        node6:http {
            address 10.0.0.6
            priority-group 5
            session monitor-enabled
            state down
        }
    }
    min-active-members 3
    monitor http
}

Yes, it is better to offline the member. Also you are correct, if you have idle connection to the servers from clients even with the pool members offline those connection will remain open until the connections timeout or the server/client issues a close/rst to the connections.