1.003 FG is way low... Ill bet fermentation was over.(EDIT okay after re reading I understand that the FG was 1.02) Do you normally just dump sugar into a bucket and bottle? I have always boiled for at least a few minutes to kill the bacteria or whatever is in the sugar, then I pour the sugar water into my bottling bucket, rack the beer, then bottle. This way I know the sugar is bacteria free and thoroughly mixed into the fermented brew.
BTW if your beer tasted sweet, you probably added too much priming sugar to some of the bottles. If you are going to go ahead and let the bottles carb up, I suggest you put them somewhere that you wouldnt mind getting beer all over(if they start exploding, it can get very messy) You could put them in a cooler, mash tun with a lid on it, bottling bucket, brew pot, etc.... I had a few bottles blow up in a closet a few years ago and it was a bitch to clean up, there was glass embedded in my walls and sticky beer in my carpet.....it sucked.