It looks like a Dracena marginata. You can cut it back if you like - wherever you cut, TWO more 'branches/arms' will grow, making it fuller.

Go ahead and let the leaves fall off, or you can simply take them all off. Give it some fresh soil, lots of natural light and let it get re-established in a pot. In the spring, when the light is more intense and the days become longer, it should perk up for you. DO NOT over-water this plant or it will rot!

To check for watering needs, stick you finger into the soil, up to about the middle knuckle - when you take your finger out of the soil, if none sticks to your finger, it's time to water - otherwise, mist the plant occasionally and give it good light and warmth.

Whether the plant lives or dies depends on several things, but the most important are A) a root system that is viable and at least a fraction of the lower stem that hasn't been too badly damaged by frost and that has a still functional vascular system B) enough energy in reserve (stored in roots and cambium) to push a new flush of leaves.

Normally, you would not want to cut a plant back in the winter, because the demand on energy reserves needed to push new foliage can easily leave the plant critically weakened and vulnerable to insects and disease, but in this case and since the foliage appears incapable of providing the plant any food (the plant's leaves are where its food comes from), there is no harm in cutting it back as hard as you see fit. Cutting it back now, will have no impact on its viability, but repotting it will. I would refrain from repotting at this point because repotting is an additional strain on the plant it really cannot afford.

The advice to be very careful not to over-water is good. The soil should feel dry deep in the pot before you water again. Increasing the humidity in the room the plant is in so it's between 40-55% would be helpful once you see signs of new growth, but misting won't do anything for the plant.

Check the plant carefully for pests - the grower might have tossed it due to an infestation of mites, mealybug, scale ..... If so, it will be important to bring the infestation under control asap.

As a precautionary measure, I would flush the soil to make sure it does not contain a high level of soluble salts. If you DO decide to take that precaution, the additional precaution of making sure you remove as much of the residual water left in the soil after flushing should be taken. If you get that far and need guidance, let me know and I'll help you.

Al

Thanks for the responses.

I haven't seen any pests on this plant (I did some looking). I'll go ahead and cut it back. After some searching, I believe that I have been overwatering it since taking it in.

The soil is very moist, so I dried out the water basin it was soaking in and I'm hoping it dries out promptyl.

Meanwhile, is there any plant food or something anyone recommends that I could give it? Help it along until spring?

No, sorry, plant fertilizer isn't medicine & should only be given when plants are in active growth (which this is not).

I'd follow the good advice suggested above & then leave it alone to try & recover. There's only so much one can do & then it becomes a matter of being patient & giving the plant time to recover.

I agree. Just leave it alone, put in a very sunny spot. It may be weeks until it needs more water, depending on the heat, humidity, and air movement in your house.

In the 3rd pic, the stems look healthy and I would not cut much off except the ruined tips until you see where the plant decides to sprout new tops. IME with these plants, there is no certainty that two tops will develop when one is cut, but it is very likely, and sometimes there are more than two.

Here is a link that might be useful: your link