OK, that's a clearer picture now. Mind you, we're diagnosing without the benefit of seeing it close up, so any suggestions made here may or may not address the problems properly. You have to decide. However....
It does sound like your problems are two-fold. As mentioned before, the basement walls and floor are losing a lot of heat to the ground. The foundation wall exposed to the air and the first foot below grade account for most of the total wall heat loss. The feel of cold air near the ceiling suggests that a major source of the discomfort (& heat loss) is air infiltration at the sill. My guess is that there is considerable stack effect in your house. On a really cold day, the outside air may be 15% heavier than the inside air. Cold air is pressing in down low and pushing lighter warm air up and out through cracks and miscellaneous openings into the attic, through electrical outlets, past poorly installed windows, etc. In winter, the inside air will become dry as a bone. Is this right?
Anyway, part of the solution is to seal off the leaks at the sill, both where the sill sits on the foundation and where the rim band around the joists sit on the sill. This is best done by a contractor with closed cell foam. You don't want to use fiberglass batts or other porous material there, because you'll have a possible moisture condensation problem in places, and that stuff won't stop air leakage. If you want to do this yourself, you can get sheet closed cell foam (XPS, or extruded styrofoam. Use a 2" thick piece or two layers of 1". Cut it for each bay, about a 1/4" narrower than the space, tack it into place up against the rim and blocked to get a small space all around the edges. After all the bays are covered with the sheet stuff, use cans of foam sealant (eg. Great Stuff), and go around all the gaps to seal the sheets into place.
You also ought to get up into the attic, lift up insulation where there are wire or light openings in the ceiling, and seal up the myriad cracks you'll find. Put sealing strips under all the electrical outlet cover plates, too. Hardware stores carry this stuff.
All this sealing of the house will radically reduce the air infiltration, make the house less drafty and cold, and cut down the heating bill. Then you can think about applying insulation to the foundation walls. Look at BuildingScience.com's web site. They have a lot of practical info on insulating basements. Plenty of good info on other sites as well.
Don't use fiberglass or other pourous batt insulation up against the basement walls. You're asking for mold and musty smell if you do that. Moisture diffuses through the wall from the ground. What you might do is apply 1" sheets of XPS foam to the concrete walls, cementing them in place with something like PL300 adhesive. You don't want an air space behind the foam sheet, unless you can totally seal the gaps around to absolutely prevent humid summer air from getting behind the foam and condensing on the cold cement wall. You also don't want a foil-faced foam or a polyethylene film over the sheet foam. You want the foam to be able to let moisture dry very slowly to the interior of the house. Of course, if you apply foam sheets, it must be covered with drywall for fire protection reasons. That's a matter of code. Leave a small gap between the drywall and the floor, so it won't wick moisture up from the slab.
You mentioned not wanting to use styrofoam on the walls, for breathing reasons. A 1" XPS layer is a vapor retarder, but not a vapor barrier. It will breathe, but at a very low rate. Various other assemblies for basement walls are described out there. Some work better than others, an some are clearly situation-dependent. You have to choose what will work best for you. It's all about movement of heat and water vapor.
There's plenty of material on all this on the internet. I've just scratched the surface for you.