A good way to become more comfortable playing two parts is to practice the right hand by itself. This way, you can develop co-ordination between the thumb and fingers without having to think about the left hand at the same time. Here are some exercises to help develop your right hand co-ordination.
It is important to be equally comfortable leading with either finger when alternating. In this example, m plays the first string and then i follows. Once you can play it this way, reverse the order. This is a good habit to develop with all pieces and exercises.