I tried using fontforge to remove unused glyphs in order to make font file size lower. I removed them from a woff file and exported as woff, but the result was low quality. Any advice?
Does it have to be a GUI tool? You may want to consider using fonttools to optimize your fonts. One tool built on fonttools that seems to match your requirements to the t is beatrix. You can give it a config that specifies which characters and font features to keep, and it will drop the rest.
I have not used beatrix myself, but I'm using fonttools to optimize woff files in production so the base is solid. If beatrix is not to your liking, there are tons of other scripts built on fonttools.
woff is probably not the best format to start from, since it's already compressed. Start, if available, with the source file (.ufo, perhaps?) If not, try .otf or .ttf. Then the main issue is probably that the export settings for hinting are different in your new version. if changing source files makes no difference, try changing those settings