I believe that when I die I will be in the same state as I was before I was born.
Quite simple really. Not scary at all. The world was here before I got here. The world will be here after I'm gone. I took part (albeit, a somewhat small part) in this world for the entire time I spent alive on this planet. Hmm... Seems fair enough.
I did find it odd / revealing the way you stated "and the fact which you're having to face and address is that you're going to die". I know you went on after that, but frankly, that really clinched my view of where you are coming from. Personally, I know that I'm gonna die. In fact, EVERY single one of is going to die. I don't find it scary or worrisome. I don't know or particularly care if it is going to happen tomorrow or a few decades. Could I miss out on a things that I'd love to see (grand kids, retirement, etc...), sure. But death is natural. Completely natural. I'd say it is typically as equally as natural as life. I say typically, because there are things that can potentially make it slightly less as natural as life (IE: Random Murder). Freak accidents are still natural, someone else cutting your life short because of an interaction (such as road rage) would still be natural (well: a natural outcome for that set of unusual circumstances, but still natural).
"How do you 'deal' with your impending death and non-existence?"
Simple: Live life while you are alive. Be sure to remember to take pleasure when you slack off. Live in a way that others would like to model (set a good example for others to follow). Don't worry much, it really doesn't help things. Be humble (Someone here has a signature that says something like this: The entire world is made up of others, except for one trifling exception).
In your scenario: A person has a failing body (a warning that death is coming), it is an event that reminds one of their mortality. Now, if you aren't aware of your mortality already the thought of that warning could be scary. But, if you are, it might be considered a blessing to know that the end is definitely near. Of course, being aware my mortality I try and be at peace with my loved ones and try to always keep in touch with those who are important but not in my daily life. Knowing that there was certainly a limited amount of time left, would let me be sure not to go without saying goodbye or having a hard feeling between me and someone I cared about. In that way, it could be a blessing.
So, I'd suggest that everyone make peace with their life while they still have one. I'm pretty comfortable with my belief that when I'm gone everything in my brain will come to a halt. All that made me who I am will no longer be. I'm fine with that. I believe it has been like that for the duration that life existed on this planet. I'm not saying that I won't live on in memories of the people I've touched while here, just that I know I will no longer 'be' when I die.