It is reassuring to work at one hand by itself while the other is not playing. The right hand is playing its small melody and the left is playing its small figuration pattern, and both are simple when separated; now they are together and nothing is familiar. This is not to say working at one hand at a time does not work; rather, it means you were working at both hands but you were treating them as separate pieces instead of parts of a whole.
When the right hand is working at its own notes, keep in mind that the left hand will be playing some notes while the right hand is playing others, and the left hand will be silent while the right hand plays still other notes. While the right hand is working, be conscious of those times the left hand is to take part in playing notes or to simply move, play a chord, play an entire measure, or simply be silent. In other words, your eyes should be on both the staves whether or not the left hand is playing or resting at that time; you must not get the notion that the right hand is playing without the left hand playing or doing anything.
The same principle applies when the left hand is alone; when it is holding a chord position or playing a bass pattern, listen to its rhythm and count with it as you play. You must imagine where the melody will fit, and your left hand must never take on a heavy, automatic sound that ignores the rhythm of the melody. It needs to be ready to share the bar.
One intermediate step is to play one hand while at the same time counting with your finger(s) the rhythm of the other hand on your lap, or tapping the beat in your left hand at the keys, or playing the right hand on the keys with your left hand not touching the keys but only counting its beats, and so on.
Another intermediate step is to combine a small part. Practice one or two measures with both hands together, as slowly as you can while not tightening your wrist or raising your shoulders. Stop before you reach the spot where the hands refuse to work together and become a tension point. If they do not work together, it is your rhythm or your counting.
Avoid playing the easy hand by itself too many times and only glancing at the difficult hand while it is doing its part. Otherwise the easier hand grows faster and louder and more impatient and difficult to coordinate later. Work on one hand at a time while keeping in mind that each hand must have an equal part to play; your right hand may have to count the note and number the fingers while the left hand has to keep quiet and maintain an even tempo. When both hands are worked on equally and then added together, it will not sound as if the right hand is now beginning at the beginning of the song.
If and when both hands finally come together in a smooth and steady flow, do not worry now about speed, dynamics or melody. Just listen to where the two hands go, to where they do not sound as if they are not the same tempo, and if you can play a note or two even when you have lost the rhythm. Working on one hand at a time is a useful practice but must always be used with the idea of working the two hands together.