Oh my, I don't even think that is a Dracaena, could it be a Cordyline terminalis(Hawwaiin Ti) ?

I didn't comment about this because I've never seen a plant with a malady that looked similar... but looking again, maybe I have? Sunburn? I wonder if maybe this plant sat outside for a while, while you were moving?

Looks like D. 'Janet Craig.'

G, how often do you water?

My JC is doing the same.. Soil got way too dry.
To be on the safe side, inspect the vase, nearest the top. Check for insects.
Dracaena's are usually pest-free, but every so often, D. Corn Plant and D. Janet Craig acquire mealy bug.

I'd cut brown leaves, then shower. Even though humidity is high, leaves need cleaning...

Thanks all!

So, it never sat outside for any length of time, but for a while it was straight in front of a window so it could have been sunburn there.

I water about once every two weeks. I will try and go check on the dryness every few days instead, do you recommend watering as soon as it gets dry on the top?

I'll also check for pests tonight, and clean the leaves.

That top stem is never going to amount to much - it may already be dead (brown or wrinkly stem would tell you). So I'd just cut it right back to the lowest Y joint and see what happens.

If it doesn't resprout, cut out entirely.

Good points, are the stems firm (and no visible pests) near the funky looking tops? Removing the damaged leaves will let you know if the problem continues. If it was sunburn, no new leaves should be affected.

Toni, sorry that happened.

Good vibes to both plants for full recovery!

The good news, 'if the stem if firm' rooting in water is simple and semi-quick.
I have a Dracaena Gold. Last spring, the stand fell to the ground, cracked the trunk.
I didn't want to toss the top, so set in a glass vase, alongside Pothos.
It's fully-rooted, ready to be potted with mom.

Tiffany, thanks..