Ask Mr. Smarty Plants

It is hard to diagnose a plant problem from a written description, but oozing sap on the trunk suggests holes in the trunk, and holes in the trunk implies some sort of pests, beetles perhaps. Have you carefully examined the trunk? You mention the lack of “red saw dust”, but is there also a lack of holes?

This link from ask extension .org talks about borers and Cherry Laurels, and may prove helpful.
The following is extracted from that link .

” Pest Infestation In addition, shot hole borers (Scolytus rugulosus) still occasionally overrun this shrub. This tiny, dark-colored beetle grows to about the size of a grain of rice and possesses strong jaws for chewing its way into the inner wood of branches and trunks. Adult females lay eggs just underneath the bark, and the larvae emerge to start mining little galleries. Signs of a shot hole borer infestation include little, round holes oozing gummy sap mixed with sawdust. Boring damage typically causes twig dieback.
Chemical control methods don't work once shot hole borers infest your shrub since the pests live in the protected area just below the bark. Preventative treatment is also difficult because the pesticide products available to home gardeners don't work effectively. Prune out and destroy all infested plant parts. If the borers attack the trunk, remove the entire shrub.

Or, your cherry laurels may be subject to feeding by the peach tree borer and/or a scale insect.”

Also helpful would be to have a knowledge person take a look at your trees to determine the cause of the problem. You might find such a person by contacting Texas AgriLife Extension.