Trying to remember this book that I read as a child (in the 80s, felt contemporary at the time). It's very fuzzy--I remember there being two plotlines that eventually intersected: one of an older woman (Millicent? Mildred? Something else entirely?) who met a dragon (Alfie?)--I believe the dragon was intelligent and could speak, and I think was killed in the process of the story; and one of a younger boy who found a dragon egg that eventually hatched, but was more like a pet lizard. It could breathe fire, though, and at one point got cut and its blood burned the boy (that's the one detail I actually remember--the rest is my best guess).
I feel like I read the book as a hardcover and it had a red binding, but chances are decent there was a book jacket that I had lost--that happened a lot. The book was in English and set in an English-speaking country. I want to say USA, but that could just be my own projection.
I have a vague memory of the adult dragon dressing in a trenchcoat to disguise himself? And hiding in a tree? He might have been on a rescue mission to recover the egg.
The book was set in the real world with no other magic or mythical creatures. I would guess it was a YA book.
I feel like the adult dragon was trying to recover the egg, but when he died the woman had to find the boy and convince him that he'd be unable to care for this dragon and he had to give him up for both their sakes.