Sounds a lot like starved glue joints. If it is climatic damage I would check every possible glue joint, bridges to soundboard, sides, block, bottom and top sills, action parts. Everything! You may wish to do this before a final commitment to the job. You may well find that putting the strings back on will be more time costly than the coast of new strings. Just handling all that chaotic mess will be real aggravating. I see no problem with your approach but I would suggest putting screws between unisons to place less stress on the bridge cat and so a later tuner can tighten the screws if they get loose. Sounds like a lot of fun. Newton
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