Shifting the focus from weight to body fat percentage

I have been diligently working on getting healthier over about the last 13 months. I am 5'9" and I started at about 255lbs., and am now down to 189.4lbs. My goal is to get down around 175, but that number is based on the flawed BMI scale. I'm getting to the point now where the remaining fat is stubborn, and I've been reading a lot about how weight training is very effective for burning fat (most of my exercise over the last year has been cardio: karate, jogging, bicycling). The problem I have with moving toward weight training is that, since muscle weighs more than fat, the scale starts moving in the wrong direction, and I'm never sure if it's added muscle or fat. I've noticed this just after a couple weeks of resistance training three times a week. I guess I realize that it is bound to happen, but it's more of a psychological thing. Has anyone dealt with this issue and have any advice for making this shift? I have a body fat monitor but it doesn't seem very accurate, so it's not a lot of help here.