Misting is not an effective way to raise the humidity around a plant unless you are in a greenhouse. The amount of time the humidity is raised is very small.
Try checking the soil and watering when the top inch is dried out. Capillary matting or a wicking system is a great way to see that the plant gets the water it needs, but not too much.
Palms are noted for getting spider mites. Check under the leaves for small white dots the size of a grain of salt. They can be controlled with soap and water applied at least three times at five to seven day intervals.
Edit: The plant in the picture is not dry but I am trying to remove all the possible problems. I suggest you remove the plastic wrap around the base.
Edit: the die back on the leaves does not look like fungal/viral activity as there are no spots or necrotic areas. It could be burn from high salt levels in the soil or shipping damage from being exposed to cold air when brought from the store.
When watering check that the entire root ball gets enough water, check for insects, place in a sunny location.
Edit: filtered tap water is fine for most tropicals. Yes you can cut off the dead part of the leaves, leave a thin margin of dead material at the end of the trimmed leaf.