I was looking back through some of the questions I'd saved as favorites and saw this answer. Looking into the trilogy it mentions, I think it is what you're looking for.
The Path trilogy by Diana Pharaoh Francis (2003-2006). The books are
- Path of Fate
- Path of Honor
- Path of Blood
The Goodreads blurb on the first book:
In the land of Kodu Riik, becoming ahalad-kaaslane is the highest honor one can achieve. It mans the Lady has chosen you to bond and be united forever with one of Her specially blessed animals -- soul to soul...
Abandoned by her parents as an infant, Reisil grew up perpetually reminded that she did not belong. There wasn't a moment when she didn't feel like apologizing just for existing -- until she became a tark and proudly discovered her talent for healing.
Now, when Reisil believes she has finally found the security and sense of belonging she has always craved, she is chosen by the Blessed Lady to be ahalad-kaaslane --one of the judges, soldiers, spies, and explorers who travel the land, bound to no one but the Lady Herself. Devastated by the thought of losing her hard-won place in the world, Reisil rejects the Lady's gift and the ahalad-kaaslane goshawk who needs her.
But Reisil's decision does not affect her alone. Soon the fate of all Kodu Riik will rest in her hands --- and Reisil will be forced to make a bitter choice to save her people from the horrors of war....
Covers from Goodreads:
Click to embiggen.
One of the Goodreads reviews is more explicit about the telepathic goshawk:
Saljane is one of the best-realized telepathic companion animals I've encountered - she provides love and support and all that wonderfulness, but she is a bird and she acts like a bird, always, not only when it would be cute or plot-convenient or funny.