I realize you don't think the book is from the Pern series, but I really think this might be Dragonsong. It is a small paperback, and part of a spinoff series, actually - which may explain why it didn't "feel" like part of the main series.
In any case, the story has a main character, a girl, who was from a sea-hold. Descriptions include fishing and gathering along beaches, for extra emphasis. There are dragons (and they hatch) in the series, but the book is more concerned with the dragons' little cousins, fire-lizards.
The main character finds a nest of the fire lizards, originally helping the queen and later returning there to wait out a dangerous storm, after running away because she wanted to be a Harper, which wasn't allowed. In any case several fire lizards (nine, not one) imprint on her when they hatch, and stay with her throughout the book as she ends up studying to be a Harper anyway.
The cover I recall is blue, with sea and sky in the background and some rock formations. The front cover has several fire-lizards surrounding a girl, the back cover does have dragons flying in.
There are a couple other covers, I added a few which have one dragon on the front, and a blue background, if one that is a better match to your description might spark your memory