This plant seems to be droopy these days, and the new shoots browned.

What is it called and does it need a lot of water?

Dracaena, I think.
When you lift the pot out of the white one, is it heavy? Does it drip? Looks very wet.

Ditto to what Sally said, the soil looks heavy, have you transplanted it lately or fertilized it?

Ditto Sally and Christine. When new growth gets discolored, it is usually a sign of root suffocation caused by soil lacking porosity, a pot without drain holes, or watering too frequently.

I think it must be watering too frequently.

As a child of a 'brown thumb's I have no idea how much to water all my green babies.

Is this more of a desert type plant? It's pretty cold down here now, so does that mean sporadic watering?

The soil is an odd kind of mulch that it came in, no transplantating or fertilising has been done, and have never seen water in the bottom but this mulch stuff seems to hold it really well and stay damp.

I've moved it to a sunnier spot, maybe that will help too? And just let it dry out?

Your plant is indoors so it doesn't matter what the temp is outside. Water it whenever the top half-inch of soil is dry, not sporadically or irregularly. If your plant is double-potted, be careful that water is not collecting between the two pots. If the pot has no drain holes, add just enough water so the top half-inch of soil dries again in about a week.

Good that you improved the light. No need to fertilize it.

Thanks WillC,

That's great, easy to follow advice!

On the temp, it's an old house (over 100 years I think!), brick, thin glass, not very well sealed up. If we're not here with the heaters on, it's basically the same temp in and out...

I do also wonder if that's hard on some of my plants? Warm > Cold > Warm > Cold...

My house doesn't run any heat or cold either, unless somebody is in that room, and is uncomfortable. Some rooms are barely above freezing by morning some days, while plants are inside for winter.

The soil looks like it might be coirpeat / cococoir. Very interesting stuff that can retain moisture for quite a while without suffocating roots, in my very limited experience with having a Sansevieria & a Dracaena in some of it, while inside over winter. (They needed bigger pots by spring.) Lifting the pot should help you gauge if it feels more lightweight than right after you watered. Lifting it out of the cache pot to water is a good idea anyway, so excess water can drip away & not get caught between the pots, as said.

Wondering if your plant might be a variegated Yucca elephantipes / gigantea. Either way, agree that more light should help get it more perky.