I'm assuming the small one on the right with the red arrow is the same as the one on the left. It looks like Butia capitata. It's possibly crown rot, where a fungus gets into the crown of the palm and rots it out. You should be able to look in there and see if there's any rot. A fungicide may be able to clear it up.

Thank you for this information. I used the palm name you gave me to do a bit more research and found the rest of my palms along with it. Apparently my palm/cycad collection includes butia capitata (pindo palms), sabal palms, and quite a few sago palms. Now I just have to figure out the best pruning practices for all. I will also be administering a broad spectrum fungicide to the crown of the palm with the collapsing new growth. Will liquid copper fungicide suffice?

I haven't had to deal with that problem but it often comes up on palm forums. From memory one of the recommended fungicides was Daconil. Copper based fungicides should also be okay, but Copper sulphate can cause problems. Read the label carefully before buying. There's no guarantee it will work but a lot of people have claimed full recovery of their plants.

I don't see cold damage on the pindo palm so best guess is that it is affected by phytophthora in the soil and roots and crown rot. Weak stems and leaflets are a classic symptom. Agri-Fos at 10 ml. per gallon as a soil drench should help revive it. The warmer the air and soil, the more effective the treatment will be. I would not remove any fronds from it unless they turn brown.

The dead, brown leaves on the palmettos can be removed. I like to leave leafs with green on them until mid-spring or later. They serve as a source for mobile, nutrient reserves when root nutrient absorption is slowed from cold temperatures.

Thank you both for the great information. I will order these products and take them with me next week when I return to the property. I will keep the post updated with recovery.