Hey Ward, here's how I do it, whether it's right or not, it works for me.
For intersecting word plaques, I usually paint the surface of the backer board flat black with spray paint, but stain would be fine. Flat black dries fast, I can usually apply two coats in an afternoon. By the next day the paint is dry enough to glue to the plaque. I use Aleen's Tacky Glue. This glue just about glues anything to anything! Before you start, get some toothpicks ready. Apply the glue to the back of the plaque. Make sure you have even coverage around the edges, but be careful around the letters. Clamp the two pieces together with plenty of clamps.
Clean up any squeeze out in the letters with the tooth picks. Let this dry for a couple of hours, then you can cut the perimeter.
Since I use flat black for the backer, applying an oil finish and topcoat won't make the backer to shiny.
You can find Aleen's Tacky Glue at any hobby store and Michael's craft store.