The more heat you add, the more humidity the room will hold. Your situation, at best, will not eliminate the moisture but you can improve the situation.
As I said, be sure the perimeter of the plastic is DOWN and up against the walls in the crawl. As long as water isn't pooling on TOP of the plastic, your humidity isn't coming from the 'dirt' in the crawl. You see, the cooler it is in the crawl, the less water the air will hold. And IF it held any moisture, the cool air would condense-out the moisture and it would pool on TOP of the plastic. Even if this is happening a little, since you have venting, it sounds like it's getting dried up.
Yes, most definitely, it's possible that your water table is high. BUT, as you can see, water is mostly present in the 'low spots'. If you can level it out, that might help 'even out' the dampness under the plastic.
No matter how hard you try to 'warm' your floor, the only way it's going to be warm is if you heat the space below, or get under-floor heating such as radiant heat systems. Remember, the floor's temperature is going to be very close to the temp. in the crawl. Thus, moisture will condense out at a lower temp. at the floor than at the ceiling. THis is probably what's causing your carpeting to fell 'damp/wet'. Insulation will help a lot so you can do as I described previously EXCEPT don't tack up plastic to the joists.
Yes, cold air will gravitate towards the warmer spots. So, in a sense, the cold air is going to 'come up and through' your floor. That is why you need to insulate. (Enough of that ...) Electric heat isn't going to "... dry everything out." Warmer air is simply going to allow more moisture to be held in it (in your case.) Just tuck that plastic down real well around the perimeter in the crawl. Some folks use gravel all along the edges; not just 'in some places'. Any exposed dirt will cause moisture to evaporate out of the dirt, into the crawl's space, and up into the house. (This is why I say TIGHT to the walls.) Temperature at the ground is cold, and in the air above it's just cool, and in the house it's just warm. So, as you can see, the air is warmer as you go higher, thus, it retains moisture.
(Now I'm just repeating myself ...) Tuck the plastic TIGHTLY along the walls, insulate as I previously stated, and fill in any 'depressions'. I have to say that since you have a perimeter drain, do you see water ever coming out it??! If not, it may be 'clogged' and NOT working. You SHOULD see something after a soaking rain. It's essentially impossible for THAT much water to be 'evaporating' in the drain pipe! You should see something. If you are, the drain is working and you have a high water table. If it isn't, then the water is 'making its way' into the crawl and simply saturating the ground, and NOT getting drained out. Fixing this is a whooooole new project.
Your post is starting to get 'buried'. If you have more ?'s, e-mail me directly. For now, my best to ya and hope this helps.