And following up on what Tom said- don't be worried about the glue joints as you do this. As long as you had good joints where the wood touched wood the glue joint will be stronger than the surrounding wood, so unless you exert force great enough to break a strip you won't break a good glue joint. If there are gaps it won't be as strong, and those will pop before the wood, but they're also easily remedied with a bit of filler and sanding after everything's back in shape.

m