I'm not excluding the possibility that you'll need some therapy to deal with your immortality... But I don't think it's terrifying. In fact people don't deal with their mortality all the time now either and most people think death is a "whole life" away from now so they mostly act and feel as if immortal today. Time will just slip by, just like now, but... forever.

And don't forget, you're just one woodworking mishap away from dying anyway, immortal as you may be