Knowing my life is finite is what makes me cherish beautiful moments. This actually often a consequence of fatal diagnoses: I could for instance observe how my father in law inceeased his interest in my children pretty much overnight after he was given one year to live by an oncologist.
There are meditation techniques focused on the awareness of death, and they can be very liberating: my favorite is called "the 9 contemplations of Atisa", if you want to look it up. I personally added a few contemplations to it, to take into account the secular setup in which we live: I find it extremely helpful to ponder the fact that we do not, and will likely never, be able to know what death is and what is the nature of consciousness. Maybe it is the end of everything, but maybe it is also the reunion with the great everything, or I will be reborn as a butterfly.
I also find the "charnel grounds" meditation to be very freeing, but some people do freak out there, so take your time.