" It’s always been in the same pot without drainage holes. "

There's your problem!! Unless a bog plant, no plants will do well in a pot without drainage. Root rots are very common when good drainage is missing. And this is especially true with cacti and succulents. They want a very fast draining potting soil that has lots of grittiness to it. Root rot is likely causing stems or branches to die off.

And if you've had it for a long time, it is likely overdue for a repot in a better soil mix a with good drainage.

When I repot it, should I cut the brown stems off? Or will they come back?

Older holiday cactus often develop thick, woody stems. I would only cut off if the stems were soft and mushy. And the roots should be firm and whitish in color as well, not brown and mushy also. Those are showing evidence of root rot and need to be removed.

From the image you provided, the symptoms predominantly suggest dehydration rather than root rot. The telltale signs of dehydration in your cactus are the shriveled appearance of the leaves and the browning of the stem tips. These symptoms generally occur when the plant is not receiving enough water or the roots are unable to effectively absorb moisture, which might initially suggest an issue with watering frequency or quantity.

However, given the context of no drainage holes in the pot, it's also wise to be cautious of potential root rot, as prolonged water retention in the soil can lead to this condition. Thiis is not usually immediately visible without removing the plant from the pot.

Dehydration,

I took the plant out of the pot and the soil was very wet and the roots were all brown. I took some cuttings and put them in water to root. I am better off putting the cuttings into soil?