Muntin
On your computer. Muntin opens LinkedIn in a real browser window docked inside the app, signed in by you, on the connection you already browse from. You can open that panel mid-campaign and watch each action happen. Your leads and messages live in a SQLite file on your disk.
The trade: nothing sends while the app is closed. Missed actions are rescheduled rather than fired in a burst, and 40% of yesterday's unused allowance carries into today.