My game is patched to 3.02.
I also have White March Part 1 and 2. I get this bug every now and then, but not sure how to reproduce the glitch. Maybe I should stop using movement speed as my rest camp bonus.
I got the same bug on my druid. I did a google search to find a solution and I came across your post.
I was able to solve my problem by adding Pillars of Eternity to Steam as a non-Steam game.
I have the GOG version.
I launched Pillars of Eternity from my Steam launcher and my main character was no longer stuck in the fast walking animation.
I hope this helps.