It -is- natural to have some fear, it is hard wired into the brain in the most primal parts. No amount of reasoning about it will make it go away. You are changing jobs [facing illness, in a burning building, etc.], and that is stressful and scary. Furthermore, not even Zazen will keep all the fear away (at least, not until we all become perfect Buddhas
). So our way (at least, in this corner of Buddhism) is not to be thoroughly without panic, fears, moments of emotional imbalance. It is to stop fighting it ... and then, though the fear might still be present to some degree, it loses its fire
Being afraid is one thing ... being 'afraid of being afraid' and resisting being afraid
is another, and pours fuel on the fire!
Often our Zazen will even let us "just sit" with the fear. And just sitting with that .... dropping resistance to the fear as well as to everything else ... can usually (not always) help that fear and anxiety lose some of its fire (because sitting resisting the fear has the effect of pouring more fuel on the fear, while sitting while not resisting the fear turns the fear into something much smaller!)
A funny thing that can result is an experience of feeling fear ... while being totally free of any fear ... ALL AT ONCE. Fear and fearlessness, as one. Most folks who have been on this Zen road for awhile will have an understanding of what I mean by that.