Your local music store may have someone who comes in (or they refer out to) for repair work. Or they may have someone in-house. One of our local ones does. In fact, my daughter's clarinet (was mine back in the day) is there right now getting checked over and repaired (failing B key and cork replacement on one piece of the barrel). Dropped it on Thursday at lunch, they called Friday just before dinner to say it is ready.
A full overhaul with all the pads being replaced, etc, may take longer. Especially the longer you wait and the closer it gets to the start of school and local marching band camps. Where we live competition marching band camp starts a week from tomorrow.
I don't know much about flutes (daughter has one of those also, for fun, but it isn't a higher end one like it sounds like you have). But I'd take that in too, and ask. I would not just assume that it'll be fine with simply replacing the pads. It might be! But depending on what got on/in it, I'd want to be sure.
If the cases got soaked (interior as well) then you will most likely want to replace all the pads on both instruments. Mold spores or mildew is not something you want to inhale when playing the instrument.
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Added: Just be sure to research the clarinet a bit before you decide to ignore it. Picked up my daughter's this morning and was chatting with the owner of the store. He had a lady come in a few weeks ago that had an old one, and her daughter wanted a newer one (an upgrade, one that was prettier with shiny keys). The one she had and wanted to get rid of was very old, but b/c of who made it and it's quality, was actually much more valuable than the newer one she was considering buying. It was a $2,000 clarinet. So be sure what you have before you decide to let it just get tossed or not seek out fixing any damage to it.
___ After your Edit:
I still would not just throw the clarinet out. It is wood, so if it has been sitting in water/moisture, it is possible that it might warp as it dries out. But just having *been* wet won't ruin it. Or if just the case was wet, it shouldn't ruin the wood. Clarinets get wet in marching band (it rains on football games) it happens. You wipe them dry with cloth. The insides of them get wet (with spit) and you use the swabs to dry them. They also should be regularly oiled to keep the actual wood from drying out and cracking. If it was well kept up, it is possible that you are only looking at pad (and maybe cork) replacement. And again, please verify the model and maker of the instrument before you assume it is now junk. Many older wooden instruments produce a different warmth of sound and are much more valuable than you might otherwise think.