I have this Underneath - it's a feeling of pure terror isn't it? Worse because you know there's nothing you can do - it's coming. I don't think about it anything like all the time but I go through periods (late night, probably stress-related) when I have to get up, make myself a hot drink and watch mindless tv/read until it dies down.
I've started trying to think it through - sort of as an unemotional, intellectual exercise, at other times when I'm not feeling the fear.
Something that I've been thinking about recently is how it doesn't faze me in the least that for the whole of history, apart from the last 37 years, I didn't exist. The world went on and I wasn't in it. Billions of years without me. And it will happen again, and really it's no different. I wasn't here. Then I was. Then I won't be.
Have you read the old Norse illustration of the concept? They liken it to a sparrow flying out of the night, through the window of a hall into the light and feasting, and back out of the opposite wall into the dark again. For such a short time the sparrow is in the light (life), the rest is unknown.
I have a tenuous religious faith but deep down I think I know that this is it. Even if there is an afterlife I believe it would be something that is so beyond that which our mortal minds can conceive that it's hardly worth trying.
Anyway, sorry - my witterings are not helpful, but you are NOT alone. I've talked to several friends about this and these feelings are really common.
Having said that, I've got a friend who has a serious anxiety disorder and is fixating on death at the moment. It's very debilitating for her as the fight-or-flight hormones that she's flooded with in response to the anxiety attacks can make it very hard for her to function while they last. She's found CBT extremely helpful in giving her tools to acknowledge these thoughts, take the power away from them and move on.
It might be worth going to the doctor if this has been going on for a while.