You are wise to recognize this moisture as a potential problem requiring a fix before you pour :-)
Whether the moisture is coming through your existing basement walls or percolating up through saturated ground, you must design a drainage system to allow this excess moisture to escape before you attemt any finish work. In my opinion, covering it with plastic is not the answer, as the moisture will remain and will eventually work its way around/over/through this barrier. Remember, hydrostatic pressure will not give up, it will not become discouraged and go someplace else (smile).
What I have done in the past is to dig a trench around the perimenter of the basement, being careful not to go below the footings. I then put down 4" drain tile (black plastic pipe with little drainage slits in it), sloping it at about 1/8 to 1/4" per foot to a low corner, where I dug a deep sump and put a sump pump. I then packed the drain pipe with small gravel. This kind of system should allow the water a path of least resistance to a low point, so you can pump it away. If the moisture is coming from a true underground aquafer or high water table (as would be the case if you were trying to build a basement in New Orleans), you would have to slope the dirt floor to the perimeter drain and then put down a thick (8 mil) plastic sheet over the dirt floor, then cover with 4" of gravel. This would force the upwelling water to the drain. Hopefully, the latter is not the source of your moisture problem.
This should be a relatively easy project for you, as you don't have to break up any concrete for your drain and it sounds like the ground is already moist and soft for easy digging :-) If you do install this kind of system, the floor dirt should gradually dry out and should stay that way, even after heavy rains and/or snow melts.
Oh, and lest I forget to mention the obvious....make sure your gutters downspouts are in good working order and drain away from the foundation and that the ground surrounding your house is pitched away from the foundation.