Ask Mr. Smarty Plants

Morella cerifera (Wax myrtle) was reported to be infected by Fusarium oxysporum or other Fusarium species in central and south Florida back in the 1990's.  Mr. Smarty Plants has found several mentions of that observation, some as recent as 2009, but no effective treatment has been reported.  The fungus often enters the plant through cut or damaged places in the root system.  It grows into the stems, blocking free passage of water and nutrients.  According to most reports, the plant eventually dies.

You may delay the demise of your plant by pruning off the dead and dying branches.  Pruning small branches in the morning would give the cut ends time to dry during the day.  Larger cut stems should be painted to prevent possible further transmission of the fungus by insects.

It may be more desirable to simply replace the plant.  If you do that and need to have the replacement in the same location, consider treating the soil with a fungicide or solarizing the soil to try and kill the Fusarium organism.  The fungicide Mycostop is recommended to protect against Fusarium, but I doubt that it would save an already infected plant.  If you can, plant the replacement in a different spot, hopefully one free of the fungus.  Make certain that the new plant that you purchase has no damage to the root system, and use only moderately enriched soil. High nitrogen encourages active growth of molds.  Make sure the new plant does not become water-stressed, since this would weaken its resistance to infection.

Wax myrtle is usually considered to be a hardy plant with few disease problems.  If you can get a healthy replacement off to a good start it should take care of itself.