The sapsuckers are drilling both for the sap, but also can come back and eat insects attracted to/trapped in the sap that oozes. Wrapping is best done in early spring and again in fall (Sept/October) to prevent further use during high-use seasons. Because additional heat damage or potential disease & pest issues are a concern with year-round wraps, you might need to either completely remove wraps over summer, or switch to a lighter mesh while it's hot out, particularly. Burlap is a good option - some use 1/4" hardware cloth or plastic mesh. 
Hanging mylar "scare" tape, or CDs, or similar shiny objects that are hung so they undulate or move unpredictably can help deter birds - until they habituate/learn to ignore them. 
Hanging suet cages away from the favorite tree(s) is a debated tactic - Some managers swear it helps, others swear it makes the problem worse. It's something to consider, and/but sapsuckers and other woodpeckers are somewhat infamous for identifying "favorite" trees or buildings...
Finally, ODFW has a brief on the woodpeckers - the red-naped sapsucker is nearly at the bottom https://myodfw.com/wildlife-viewing/species/kingfishers-and-woodpeckers
And - all native bird species are protected, so our solutions need to be non-lethal.