@Lydia Kay
This all sounds scary and overwhelming but these are the sorts of things you should be prepared for in real estate. Here's how I would approach it:
1) install new splashblock or diverter at the bottom of the downspout and make sure the water is being diverted away from the house from now on. Make sure any other downspouts that go into the ground are free and clear. Usually they are not. Often you are better off just diverting the water away from the house with a splash block. Make sure all of the soil is graded away from the house. If it's not, you probably need a french drain and some catch basins.
2) put a dehumidifier in the damaged area. Run it for days/weeks or whatever. Dry it out as best as possible.
3) depending how much rot there is, some you may need to pull out and replace and some you might not. You might be able to get away with just sistering some joists in certain places.
4) fix the floor and continue your remodel.
Could you take a couple of pictures and post them here? Is there a sufficient crawlspace?
You should call up some handymen and carpenters in your area and get bids. This should NOT be insanely expensive. By your description, it all sounds pretty fixable but it's hard to tell without seeing pictures of it. Is it a beam that is water damaged or is it a sill plate that sits on top of the foundation wall?
Try not to get too discouraged about this. First thing right away though is to fix up those downspouts and make sure any water intrusion stops asap.