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This could be a watering problem, either over- or underwatering depending on whether the soil is drying out enough between waterings. A plant with few leaves will not use very much water and could develop root damage if the roots stay in wet soil too long. Also, an underwatered plant with wilting leaves will perk up after watering, while an overwatered plant will not.

https://www.gardeningknowhow.com/ornamental/flowers/plumeria/growing-plumeria.htm

There is a chance that the other plant was killed by a disease. In case that's true, make sure the soil and any plant material from the other plant has been removed from the area and any gardening tools used on the other plant have been sterilized with a 10% bleach solution.

Plumeria is hardy in zones 9 through 11. If you have planted your Plumeria into the ground, make sure you are in these zones.
If you are in a colder zone you will need to keep a Plumeria in a container, to winter it over in an indoor location.

Plumeria roots are shallow and fibrous, they should not be a problem in the ground.
Here is an article with care information.

https://www.gardeningknowhow.com/ornamental/flowers/plumeria/growing-plumeria.htm

Wait until spring (after the last chance of frost in your region) before pruning to avoid any additional frost damage. Once it's spring, you can prune the tips back, removing the damaged portion until you see healthy tissue inside the branches:

https://www.gardeningknowhow.com/plant-problems/environmental/tips-for-saving-cold-damaged-plants.htm
http://msue.anr.msu.edu/news/pruning_frost_damaged_plants

You should remove the leaves on the bottom half of the cutting. The leaves on the top half can be left on, or you can remove all but the top pair of leaves.

From my research the stems of Plumeria splitting in some fashion is normal.
It can be caused by rapid growth, excess watering, sun scald or exposure to extreme temperatures. It also can be more common in certain varieties then others.

Unless you see soft spots on the trunk, there is no concern.

https://www.gardeningknowhow.com/ornamental/flowers/plumeria/growing-plumeria.htm

Here is a link with information on care and pruning.

https://www.gardeningknowhow.com/ornamental/flowers/plumeria/plumeria-pruning-info.htm