My A/C is emitting this dank(?) smell, but only when the cooling is turned on, but not the heat. What part of my system would be the source of the odor and is this something I can remedy myself or is a service call required?

Maybe you could clean the filter in warm soapy water and then when dry, dab it with ONE DROP of vanilla essence

When you use your airconditioner it takes water out of the air and it drips into a pan and is either drained to the outside or down you sewer line. This pan can get very filthy form all of the stuff that gets thro9ugh the filter and sticks to the coil and then drops into the pan. After a while this becomes a very stagnant and smelly place and the air going through the coil carries the odor with it.

In the winter the water in this pan is al dry from not using the airconditioner and you have started using the furnace which drys the stuff in the pan and therefore the odor goes away.

You can atempt to get the front off of your coil and see if you can clean some of the stuff out.

I service man can take the coil out and have it steam cleaned but it is not cheap because it takes several hours to pump the freon down, cut the lines, seal them, get the coil cleaned and then reverse the procedure to get it all back together.