I'm trying to identity a book series (I think it was 3 or 4 books) that I read in the late 90s or early 00s. I'm not sure if it was a newer series or not. It had a distinct "coming of age" element so may have been a YA series. Many specifics below are fuzzy but hopefully the overall plot details will be enough (e.g., there are lots of spoilers ahead)
The main character (a woman) and her male companion (who may have just been a friend initially but they were married by the end) arrive at this kinda-monastery place for reasons I don't remember. The woman ends up learning magic there from some man that is initially in a jail cell. I think this used to be a center of magic training but the "magicians" are a dying breed and her teacher was effectively abandoned in jail, so there is no one else around.
Her teacher, like all magic users, is also a shape shifter. However there are only a few creatures they can become safely become and one of them is a large flying creature (think dragon, although they are called something else – maybe a wyvern?). However, turning into a dragon isn't just something they "do". Their dragon form is a key part of who they are – they are as much "dragon" as they are human.
The main character doesn't understand this and it is initially hidden from her by her magic teacher because she has to somehow figure this out on her own. I think it is the end of the first book when she transforms for the first time. It's apparently a very dangerous process because it runs the risk of having your "dragon side" take over and effectively becoming a wild animal.
Normally these magic users are trained to "destroy" the dragon half of their personality but she refuses and makes a "pact" with her dragon half, effectively spending most of the book series with split personalities.
I don't remember most of the book series, although I'm pretty sure one of them revolves around time travel (where she gets a glimpse of life a long time ago when the magic users were still at their prime).
I'm pretty sure the last book involves her finding the last "colony" of the magic users who are living in an island in the middle of the ocean. The leader of the island faction may have been the wife of the man left in jail, who became the main character's teacher. The main character is not very welcome there because of her unusual background and training. In addition, the leader has a very domineering personality and of course the main character has a strong independent streak. Finally, her love interest is not approved of because it turns out that being a magic user is the result of a rare recessive gene, and so mates are assigned to improve the odds that children will also be magic users.
Things get really bad when the leader of the island faction discovers that the main character didn't destroy the dragon "half" of her personality but instead made a pact with it. This is considered extremely dangerous due to the risk of going "wild" (which is a theme through the entire book series).
Rather than destroying the dragon half of her personality, the main character merges the two halves of her personality, becoming both a more complete human and dragon. It is revealed that her dragon half wasn't something "external" but just represented parts of her personality that she had before she first transformed. It is further revealed that this insistence on destroying the dragon half is the wrong choice, and leaves them (in a sense) weakened and incomplete, leading to their slow decline overall.
Finally, it turns out that her love interest has a single copy of the recessive gene (which gives him a less useful set of special abilities – I think he can see ghosts) and so she is given permission to be with him.