A "bias toward the truth." BWAHAHA!!! That is about as non-introspective as I've ever heard.

Sort of like "my greatest fault is that I'm just so honest and humble and kind that I'm sometimes my own worst enemy."
I recognize that I have a strong affinity for, and confirmation bias toward, weirdness of all varieties. To some extent, I share the conspiracy-prone mindset. This cuts across all varieties of weirdness in which I have been heavily involved - religion, UFOs, psychical research, the JFKA and numerous others.
The only thing I do to stay on the side of rationality is to try to be relentlessly critical and skeptical. I am the 180-degree opposite of the Gee Whiz True Believer in every area. This is true even of my own paranormal experiences. My first reaction to every super-duper UFO tale or Near-Death Experience is "Bullsh*t."
That's all I know to do - recognize the direction in which your confirmation biases point and then be relentlessly critical and skeptical of everything that feeds into them. When a UFO case or Near-Death Experience or other Tale of Weirdness now survives my filter - and some do - I am satisfied it's a piece of evidence that is worthy of being factored into my belief system.
The other danger is being so aware of your confirmation biases and so viligant that this becomes a confirmation bias of its own - because by God you aren't going to fall prey to your confirmation biases, you swing too far in the other direction.
I was on a few disciplinary panels for other lawyers. My biases tended to be personal - I either liked the attorney on trial and felt affinity or sympathy or didn't like him or her and felt the opposite. Here as well, all I could do was try be honest with myself and not let this bias affect my evaluation of the evidence or the discipline too much. Also not to let my role as a judge lure me into playing ego/power games. I always tried to put myself in the attorney's shoes and err on the side of compassion if I reasonably could.