that the heat had something to do with shrinking and caused it to buckle.
Yes of course you can use liquid nails for paneling, BUT if you ever want to remove the wainscoting, you have a big problem with the glue and what damage it does to the drywall.
But if that is not an issue, use this glue in squiggly figure "S" and then also use paneling nails and locate studs initially so that nails also go into them.
This should work. Also when glue installed, roll the wainscoting out so that all the glue hits the wall surfaces. If there are inverted bows in the walls, then apply nails there to have wainscoting attach there also.