I too, place virtually all my reservations
online now days for the mileage bonuses. (Beside, it's fun not to have to talk to anyone! )
But I found a great way to get through to the airlines when all the lines are busy. This could be due, for example, to bad weather when numerous flights get cancelled and everyone is trying to get through.
What I do is call the number for Spanish speaking people. It's usually much easier to get through. The agents, of course, speak English as well and aren't about to hang up on me just because I can't speak Spanish.
Neal