With locking tuners, you shouldn't ever have to have a full turn of the string (or more) round the post. The idea of them is to get rid of as much excess string length as possible. So when putting the strings on, get the post angled so that the string goes straight through it when fed from the nut, pull it tight and lock it down. You should then be able to get up to tune within 1/2 to 3/4 of a full turn of the post. You haven't got a trem, so it's not as critical, but if you had, on a standard tuner, the relaxing of tension when using the trem allows the normal 2 or 3 turns to slacken off a bit and reposition themselves. When the string comes back to normal tension those turns can then move a bit afterwards, normally to contract a bit more, and the tuning goes out.
The main idea of the locking tuner is to remove those (now unnecessary) string turns and so to remove one cause of guitar out-of-tuneness.