I remember reading a book but at the time couldn't find the next in the series and never got back to it. The book is at least 15 years old and there is a minimum of 3 books in the series. It had two separate plots, one about dragon shifters. These shifters were born to humans and then learned they could shift to dragons later in life. They lived apart from the main society and were paired off (every dragon born had a mate). The other plot line was about the royal family, specifically the king's main concubine who was cheating on the king with his bastard brother. I feel like they may have been called the pheonix king line but I'm not sure. There was a prophecy concerning her child. Any help would be greatly appreciated!
That is The Last Dragonlord by Joanne Bertin. Published in 1998 with the sequel in 1999, the trilogy wasn't completed until 2012. Like you, though, I feel it was clearly stated to be 3 books.
Dragonlord Linden Rathan, last-born of a race of immortal weredragons, has spent six hundred years alone, searching for his soultwin while his fellow Dragonlords watch over humanity's Five Kingdoms.
When the Queen of Cassori dies mysteriously, Linden and the other Dragonlords are called upon to prevent civil war as two human claimants vie for the regency.
As the battle for Cassori rule escalates, Linden becomes the target of the Fellowship, a secret society of true-humans who could actually destroy his immortal life.
Then he meets a beautiful young ship captain named Maurynna who may be the only one who can help Linden bring Cassori back from the brink of chaos.
I don't recall a royal line called phoenixes, but it's all pretty vague. However, the second book IS titlted Dragon and Phoenix.
This Goodreads review has additional details in its summary (edited for content):
The Last Dragonlord follows a lot of characters and is told from multiple point of views, but we can say that the focus is on Linden Rathan. Linden is a Dragonlord, he can shift from human to dragon form (and vice versa of course). In fact, Linden is the last Dragonlord - after his Change no more potential Dragonlords were born. It's a bummer for multiple reasons:
All other Dragonlords call him the "little one" - endearment for youngest in their group. Since Linden is the tallest and biggest among then and more than 600 years old, this is really annoying.
Linden did not meet his soultwin yet - the female Dragonlord that is his mate, the other half of his soul, well his soultwin. She still needs to be born/changed and after so many years waiting, Linden has lost almost all hope that he will ever find true love.
The plot of the novel revolves around court intrigues in Kindgom of Cassori where a regent needs to be chosen to rule until young prince grows up. Since Dragonlords are "ancient Givers of Law" Linden and two of his friends come to judge who has the most rights to assume this role. Although almost everyone in this novel is plotting something, there is little tension and The Last Dragonlord, although not exactly a boring read, is also not a page-turner. I think the problem is that Joanne Bertin covered all view-points so we always know what all sides are plotting.
The romance in the book is believe it or not a love triangle. We will have Sherrine, beautiful noble lady, and Maurynna, trade-ship captain, fighting for Linden's attention. But don't expect much tension here, because Linden tell us immediately who he is going to pick, so the whole love-triangle business is more in the head of the rejected woman than it's really happening.