This will sound stupid, but use less glue and spread it with your finger tip. Keeping the inside of the box clean is more difficult than cleaning up the outside, so after I apply glue to the entire surface of the linings I push most of the glue off the inside edge to minimize squeeze out inside the box. I have used solid linings on the last several so no kerfs to leak glue onto the sides. I like to have a thinner application over the whole surface rather than a bead on the edge and hope it squeezes out over the entire surface.
I am not known for cosmetically perfect interiors, and would rather have a little squeeze out showing than a loose brace. I try not to be too sloppy, but sometimes it appears that way.
