The experience of being a Nazgul is that it doesn't matter.
It's not that you aren't important. You know perfectly well you're a terrifying death monster whose mere presence drives foes before you or whatever. And it's not that the things you're doing don't matter. Foes, driving, etc. But you are very aware that your experience of it doesn't matter.
The very first thing the Ring does is let him see your thoughts. After however long you spend resisting, or even during it, you will get used to filtering everything: this thought starts a fight, that one is safe. It is very easy for humans to build habits, that way.
Your first reaction to everything will always be about what he would think of it. Maybe you used to have hobbies or enjoy music. Now that's a waste of time. Or maybe you used to think the all-black color scheme was trying too hard. You're still going to Rohan to kidnap specifically those horses. You will think this way, in your own head, forever.
Eventually it's just-- whatever he says, that's what's happening now. You aren't really a participant. You're barely an observer. If he ever gets angry at you for failing to show initiative, well, apparently that's what's happening too.
You're still in there somewhere, feeling things, but what one might call "you" doesn't really connect to what you do or say or think. You ever feel like death would be better than wraithdom? Stop that; you have a task to do and now you're bringing down morale for everyone else. (Was that your voice or his? Does it matter?)
Every time you do anything, you know in advance that you won't be good enough. Maybe some of the Nine are better at things than you, but that too doesn't matter. His standards would be perfection either way, whether or not the sentence "Khamul would have done that right" features.
He himself, obviously, would always have done it right. He's smarter and better at everything than you. Or at least everything that still matters.