I would cut it back below the point where it branches (don't worry: if/when it comes back, it will sprout from more than one place). If the base of the plant is still solid, there's every reason to think it can still come back. I'm not sure whether the mushiness would spread if the plant were left to its own devices or not: I can argue it both ways to myself, and both arguments sound really convincing. So I don't know. If the stem is firm for most of its length, the plant isn't dead. As far as the root check: there's really not that much of a trick to it; just pull the rootball out of the pot and remove (= cut off with a clean, sharp knife) any roots that look dark, feel mushy, or smell rotten. A normal, healthy Dracaena root will be white to orange in color and firm. Be sure to cut away all of the damaged tissue, even if that means cutting off a little of the healthy stuff too. Telling you earlier not to up-pot the plant was more because up-potting will cause the soil to retain even more water over an even longer period, which is exactly what you do not want. It's not that the roots are particularly delicate. After you're done, put the plant in new, quick-draining soil, in the smallest pot the roots will fit into (it looks like yours is in a 6-inch pot now; if you can get the roots into a 4- or 5-inch pot, go ahead, and if not then clean the same pot with soap and water and re-use it). You don't necessarily need to water it immediately. Give the plant a few days before watering, and leave it in a bright (full or filtered sun in a south window is good) warm (75-90F) spot and be very patient. Check the stem occasionally to make sure that it's not going mushy. You shouldn't need to water much during this time, but it's okay to water when the soil becomes completely dry. (Never ever ever ever ever ever ever ever ever ever let this one stand in its drainage water for more than about fifteen minutes.) If it intends to come back, you should see little buds forming at the top of the cane within maybe 6-8 weeks. When that happens, you can move it to a slightly darker location, but still water cautiously until it's back in full leaf again, because it will be using water very slowly for a while.

Well, what do you mean by not much light... maybe not full sun, but it will need some light. When you water, is the soil totally dry? The water may be running down the dry soil on the outside of the soil between the pot and soil OR there is no drainage and that's too much water and it's rotting. Those are my two guesses.

How do you water your plant? Do you saturate the soil or just give it sips? Do you allow the water to run off after watering? When was the last time you took it to the shower to get a good soak? Fluoride does cause burned tips BUT if you are not properly watering it could mean it's getting burned from salts in the soil already. Take it to the shower and give it a good drench, then return it. Cut off the brown tips at an angle because they are not pretty but do not cut off the whole leave. The plant will withdraw the nutrients from the leaves as they dry and die. This is a normal wintertime activity because it also means the plant is not getting enough light and it is slowing down. I'd also move it closer to a south facing window, if you don't have a south facing window, out it in front of a West or East window. Not to the side of the window, in front of. See if that doesn't help your plant.

I can't say for sure if its under watered but you should always water thoroughly when you water your plant. It flushes salts out the soil. Don't be scared to give your plant a good drink at watering time. And instead of watering every two weeks water when the soil feels dry a few inches down. Your plant does look a bit thirsty. What type of soil is it in?