Sounds as if your transom core material is most probably bad, and needs replacing. Not an easy job, but necessary if you plan on keeping boat.
First you need to find the extent of the problem. Do this by removing hardware attaching screws and checking for moisture, drilling into core in various areas sticking in "ice pick" type tools to feel for soft wood. Good areas to start checking are, first and foremost, the infamous "water trap" aluminum angle molding lining exterior edge of transom notch. remove screws and push awl or ice pick into screw hole and see if wood is soft. Next remove screws holding transducers, trim tabs, speedo wheels, on lower exterior transom. If transom core is saturated, water will come out of these screw holes.
If you detect signs of water while doing these checks, you'll need to check even further to see if complete core is compromised. I'll tell you from experience, if you find wet wood in the screw holes holding aluminum angle molding in transom notch area, AND, water comes out from screw holes on bottom of exterior transom, then you need to replace your complete transom core. That is especially true for your boat, since you already state that area around your thru hulls is soft.
It you do not have any attaching screw holes near the bottom of your outside transom area, then drill three or four 1/8" drill holes in lower exterior transom around an 1 1/2" up from lower edge, spacing them every few feet from side to side. Drill these holes only around 1" deep into transom. Don't worry about repairing these holes, they are easily patched with an epoxy/fiberglass flour mix.
Now you should know if your transom is good or bad, If you found no traces of moisture, then perhaps you only have bad wood around thruhullls, and you may be able to remove bad wood in these areas, completely dry, and then apply a "SeaCast" type of curing mixture. If you found water dripping from lower exterior transom holes, then your transom core is probably completely bad. Believe me, it is a lot more common than most owners are aware of. Below is link to a "step-by-step" transom rebuild I did on my mid-eighties Seafarer, after I found that core was shot. The first three pages pretty much describe the complete rebuild, and after that it is more of comments from others. It will let you know what is involved, and what procedures you can follow.
http://forums.bateau2.com/viewtopic.php?f=10&t=19949
Again, this is not easy, and not for the faint of heart, because it requires cutting out and replacing structural sections of your boat, but then again, it isn't rocket science either. If you decide to try and do this repair and need any info, feel free to PM me. Good luck, Mike.