Water IS getting to it. There IS water there. You cannot fix it by covering the wet. You must fix it by stopping the water from getting to that point.
Check your gutters. Do they tend to get clogged? Are the downspouts set so the water is directed away from that area?
Check the grading in that area.
If all of this checks out, then the water is getting through the wall. There may be a crack. You may need to dig down and apply water proofing in that area, or repair the crack.
This is going to be sort of a comment, or at least not a complete answer.
When designing the system, the goal is not to create a breeze through the space. The goal is to control pressure equalization in such a way that the air is mixed and dehumidified (conditioned).
Having one large return at one end of the space, and a large return at the other may work. But it's not going to be as efficient, and will require maintaining an open area for which the air to flow (no closed doors).
Let's back up a bit, and think about how the system works...
The fan (or blower) is going to create a high pressure area "in front" of it, which causes the pressure in the supply ducts to increase. The higher pressure air flows from the supply ducts, in an attempt to equalize the pressure between the supply ducts and the rooms. At the same time, the fan (blower) has also created a low pressure zone "behind" it. The higher pressure air in the rooms is forced into the return ducts, in an effort to equalize the pressure between the rooms and the return ducts.
So the system works by forcing conditioned air into the rooms, and having unconditioned air pushed into the system.
Wherever you have a supply duct, you want to have and unobstructed return duct. Otherwise you'll end up with a high pressure area in one place, and a low pressure area in another.
In your example, if you have a single supply duct feeding the kitchen, and a single return in the bedroom. What happens if the bedroom door is closed? In fact, the bedroom door in your drawing would have a tendency to be pushed closed, if you installed the system the way you've described. You'll be creating a high pressure zone in the kitchen and living room, and a low pressure area in the bedroom. Which means the door could be closed by the pressure differences.
You'll want to install a supply and return in each separate area. In your case that would mean installing a supply and return in the living room/kitchen, bedroom, and storage area.
Unfortunately, duct design is a bit of science, and a bit of art. Covering all the information you'd need to know, would be nearly impossible in this short answer format. In my opinion, duct design is not a job for an amateur. At the very least, you'll want a pro to assist in that aspect of the job.
I know I've rambled a bit, and I'm sure you don't want somebody to tell you not to do it yourself. But in my opinion, it's better to have it done right, then to have it done cheap. Aside from all that I've mentioned. It's impossible for somebody to design a duct system without knowing the equipment you're going to use, or the actual dimensions of the space. So there's no way anybody on this site could provide a duct design for you, with the information you've provided.
My advice... Get an HVAC contractor to design the system for you, then you can do the work of installing it yourself.
Best Answer
You will be bound by building code - e.g., minimum number and location of electrical receptacles, minimum size of doorway, etc. But definitely not a problem finishing only part of the basement. That is actually quite common - no reason to spend money making finished space if it is going to be used for storage, workshop, laundry room, etc.