And in general. The old adage about lousy food and terrible portions rings true for me nowadays. One part of me wants to disappear into oblivion right now and another is saddened we can’t all live for much, much longer.
Mortality is a simple fact. We’re all going to die someday, and that’s that. One life, then it’s over. It’s not mortality that needs coping with, it’s all the crap we have to deal with here and now.
I like to call it living in the moment, but denial might be a better word. Honestly, I don’t think about it much.
I pretend I’ll be young and invincible forever. The twinges in my back and knees make that a little harder than it was 10 years ago, of course. 
But really, aside from denial… sometimes when I’m hanging out or cuddling with my boyfriend, I have a moment where I think, “If I died now, at least I’d die happy.” I just try to maximize the number of those moments.
Robert Ringer, author of Winning Through Intimidation and Looking Out For #1 (the second book largely repudiates what he wrote in the first book), offers a philosophy he calls “The Iceball Theorem”:
[quoted from memory, so probably not his precise wording]
A hundred billion [or whatever] years from now, when the Sun is a burned-out shell and Earth is nothing more than a cold frozen ice ball, none of this will matter.
[/quoted from memory, so probably not his precise wording]
ETA: His second book (mentioned above) is largely similar (even nearly a clone in some parts) of Harry Browne’s How I Found Freedom In An Unfree World.
I cope with it by trying to live as much as I can right now - because who knows when Time is going to force me to give up doing the things I now enjoy?
A hundred years ago, I wasn’t. This wasn’t much of an inconvenience. A hundred years from now, I won’t be. I doubt that’ll bother me all that much, either.
For me, accepting my mortality went hand-in-hand with accepting my insignificance in the universe. Think of the billions of other stars out there and the likelihood of life throughout the universe, and think of every creature on this planet that lives now or has ever lived. Despite not really being able to grasp the enormity of that, you gain perspective. In the grand scheme of things, I’m nothing. I and everything I know will eventually be gone and forgotten.
I don’t want to die young and I don’t want to die traumatically, but death is part of life. I’m one of countless organisms that will someday complete the cycle.
Speaking as someone who had an intense bout of depression based on the same issues the OP mentioned? That philosophy makes it worse.
I’m optimistic enough to think that maybe this will matter. Maybe we’ll get off this rock and over to another one and keep on going.
To answer the OP’s question: I look at my two sons. I know that what I do in my time here will live on through them, and through their children, and so on.
I deal with it by embracing low self-esteem. I consider my mortality the one true blessing I can confer on the world. Sure, someone will have to haul off my carcass, but after that nobody will have to put up with me anymore.
It’s a comforting philosophy in the right context. In terms of the choices (and mistakes) I make, my anxiety level goes DOWN when I remind myself that nothing I do will matter in 100 years. Way down!
However, in terms of mortality, I agree with you.
I’m more curious about dying than I am afraid. For one thing, I don’t think it will hurt. I may be in a lot of pain beforehand but I think, in that case, death would be a release.
I’ve been really, really sick before. I’ve had my body mutiny on me in a sudden onslaught. (One minute you’re fine, the next minute you’re gasping like a fish out of water.) During those times I’ve experienced a withdrawal, a hyper-awareness of not reality, but a deeper me. “Fireworks” moments of understanding my connection in the world. So I’m kind of looking forward to the revelations in the big sleep. But with all the questions I have ready I suspect the answer felt might be “What does it matter?”
I don’t think we go to heaven and I don’t think we cease to exist. I think we disperse, molecules ready to come together again. Like it was before we were born.
Enjoy the now. Nothing else matters.
I also find comfort in the thought of how insignificant I am in the context of the universe. And the idea that we’re all made of the same molecules, over and over again, makes me feel more philosophical about the fact that the specific ones I’m composed of will cease to cohere one of these days.
Aside from that, the idea of my own death doesn’t bother me. Because I don’t believe in any kind of literal afterlife, my death means the end of perception for me, and once perception ends, there are really no worries. It’s the thought of my loved ones’ mortality that really gets to me.
It’s not my mortality that bothers me. I won’t experience my death. When I’m dead, I won’t exist and I won’t experience anything. To me, worrying about my non-existence after I die is as silly as worrying about my non-existence before I was born.
What does bother me is the mortality of other people. When friends and family members die, that’s something I have to live through. Those are the deaths I worry about.
Music. Drugs. Good Friends. Solitary walks with a dog and a book. Shit happens, might as well be there.
Why it’s useless to commit suicide, no matter how bad things are:
(Assuming there is no consciousness in any kind of after-life.)
If you successfully kill yourself to escape the horribleness of your life, you will never know that you succeeded!
Now how useless is that?
ETA:
Similarly, the question sometimes comes up (it was mentioned in a thread about the Large Hadron Collider a while back): Is there some danger that it will create a mini black hole that will absorb more and more stuff until the whole Earth, the whole Solar System, the whole Galaxy, and more, is destroyed? And will it happen (at least the destruction of the Earth) so fast that we will never even know it?
I pointed out this: If this would destroy the Earth so fast that we will never even know it, then by a certain extension of the logic, we would also never know if that hasn’t happened. So, how can we be sure, even now, that the Earth hasn’t already been destroyed by a black hold from the Large Hadron Collider?
Wait. What?
There’s a pretty immense hole in that logic. Yes, we wouldn’t know if the world was destroyed because we’d be dead. But we’re not dead. And we can experience life and the world and know from that experience that we’re alive and the world is still here.
I cope with my mortality by not dying. One day I won’t be able cope anymore.