Hello Jim and welcome. My name is ***** ***** I'm a biologist with a special interest in tree growing. I'm sorry to hear of this incident.
The only way to know for sure is to remove the palms from their pots, and remove the soil from around the roots. Do this on newspaper or something else you can wrap up and dispose of in a trash bag. Root rot is caused by a fungus and you don't want to harbor it in your house or yard. Once the palms are laid out, examine the roots. Healthy roots will be white and firm. Rotted roots will be dark and somewhat soft. If all the roots on a particular palm are unhealthy, you won't be able to save that one. When there are both healthy and rotted roots, use pruning shears to cut away all the rotted roots. When you're finished, disinfect the shears in a solution of water and household bleach.
When all the bad roots are trimmed off, you're ready to replant. You'll need to use new pots, that are only slightly bigger in diameter than what remains of the roots. Use a sterile potting mix. Place the roots in a hole in the center of the potting mix. Next, mix one teaspoon of thiophanate-methyl per gallon of water and pour it over the roots. Thiophanate-methyl or a similar fungicide should be available in a nursery of garden center. Finally, cover the roots with the potting mix.
For the first two months in the new pots, pour a fungicide drench (water and fungicide) into the pot to control and fungus still remaining. Resume your regular care. This will be all you can do. Then it's up to the palms.
If you have more questions, let me know in a REPLY. I wish you success in saving your palms.
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