Try lighter fluid. I have used lighter fluid to remove gum from sweaters, jeans, jackets with no residue or problem whatsoever. Dab it on gently and it usually dissolves the gum.

I have washed 100% wool and 100% cashmeres in the washing machine..! I use cold water, mild detergent and use the delicate shortest cycle. Remove carefully from washer and I hang by the armholes, reshape if necessary, over the towel bar, or shower curtain bar to air dry. - Takes about 2 days to fully dry. I never experienced any shrinkage. Never put your sweaters on a hangar...when wet or dry...