Terry flats are HUGE on a newborn. I've seen some mums cut them in half to use on newborns or you can use flannelette flats or cut up a cotton baby blanket/sheet.
We didn't use covers at home. Changed as soon as bub was wet, keeps rashes away.
Started with snappis but then switched to velcro belts...
And we didn't soak, we just did a rinse before the main wash.
Oh and you can get cheap 2nd hand covers HERE: