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There is a good chance that either you have a roof leak ( are these at the bottom of a valley perhaps?) or there is not adequate overhang or metal drip edge to shake the water off and it is bleeding down over the tails and wicking up into them.
In either instance, it would appear that some removal of roof is needed to effect that repair. At that time, putting in a dutchman scab is fairly easy. The thing you will need to watch out for is if the roof was framed with trusses. These things are delicate when you start taking them apart and really need expert framing people to do.