Bear, when plants show signs of rotting, immediately one thinks of "overwtering". Cactus -- I think pretty well all--are succulents and therefore, they don't need as much watering as other plants; the leaves store moisture longer.

OK...aside from that, you say you don't think you have given it too much. In winter of course, the plant, not growing, needs that much less watering. There is less sunlight to cause it to grow and use the water.

How does your pot drain? Cactus are usually planted in soil that drains very well. It's mix can contain material that lends itself to draining well.....peat moss, soiless mix, vermiculite...

If it is not too much a bother, why not unpot your plant; take a look at the roots, see if any smell comes from them or any sign of rotting.
If so, cut that away....tear your roots free of any compacted soil and maybe --as a last resort though, give it fresh potting soil. Usually giving a plant fresh soil can bring about fresh growth...that you don't want..not right now. Do that round about mid February.
If rotting is suspected, you might at the plant's next watering, give it a few drops of "Damp Off"...a fungicide which will kill any bacteria that maybe is in the present soil and that which might be clinging to the roots.

See that the plant does drain well and let it dry some between waterings. Remember though, it is a succulent and can go dry a bit more.

The winter solstace is approaching...December 21..the shortest day of the year so don't be afraid to give your cactus some better sun. (south, west or in a pinch an east exposure)

Don't feed it. Feeding only prompts the plant to grow...and at this time of season, it wont do that.
The food would be wasted and dangerous salt build-up could result.

I suggest for water, leave it out overnight to gain room temperature---don't give it cold water.
If you like, put the water on top of the refrigerator and let it gain a little warmth from that position.