Both books came out in the last 10-15 years or so (late 90's early 2000's) and both were written by the same female author (this is why I've listed both here under a single question).
The first book (told in first person) is about Gilgamesh (the name Gilgamesh is not in the book title) and he tells his story from when he was a child far in the past and continues to the present day and then into the far future when the earth has been swallowed by the sun and he is just a disembodied mind.
The second is about Lilith (name Lilith is again not in the book title) and on this one I'm much more vague on the overall story, sorry. I believe that this one is also told in the first person.
From what I remember, the books aren't connected as part of a series, they are both single self contained novels.
I got both books from the library (I read the Gilgamesh one first and liked it, so I looked for more by that author and found the Lilith one) in the past 10 years and neither book was very old when I got them. A search online just gets me books with Gilgamesh and Lilith in the titles.