The best way to avoid looking at the keys, and often the easiest way to play the keyboard, is to walk the fingers along the keys. In other words, by walking the fingers, or in other words stretching/reaching with the fingers to find the note, you maintain more contact with the keys. As opposed to lifting the hand from the keyboard and planting it somewhere else based primarily on sight. In other words, let your fingers take bigger steps.
To illustrate what I'm talking about, I came across this situation in the folk song
Tum Balalaika, in which the following broken up chords are played in the bass
cleff in the first two measures. BTW, the second A is an octave to the left
of the first A :
D, F, A, A, C#, E
Different ways of playing this, with identical fingering:
5, 3, 1,(hold thumb on note) ...stretching 5, 3, 1
I could also play it with the same fingering, but non legato without gliding across the keyboard, or stretching the 5th finger. Which would be
like this:
5,3,1 ( and letting go of note) ..move hand to the left.. 5,3,1
Which way helps you keep more contact with the keys? You would be playing more by feel in the first example, while the 2nd example you'd be playing more by sight.