I don't see the fascination with wood glue-- for bonding paper to balsa white glue is at least as strong if not stronger.
There shouldn't be a mess if done correctly. It only takes a little glue spread VERY VERY thin to bond printer paper to the average size fin... I usually use a drop about 1/4 to about 3/8 inch across to bond paper to one side of a fin about the size of a Mean Machine or Big Bertha fin, which is probably what I'd consider an "average" size rocket fin...
I cut my fins, stack sand them, sand in any desired airfoil, usually round leading edge teardrop aft edge (depending on the rocket). Grab a couple sheets of paper out of the printer. Lay the fin with the leading edge facing the center of the paper, about an inch from the edge of the paper, trace it lightly with a pencil, roll the fin over the leading edge (making sure it doesn't slip) and trace it lightly again. Put the drop of glue in the center of the outlines, and spread it with a finger COMPLETELY over the entire outlined surface area, and extending a bit past it all the way around, place the fin down on the outline, and press firmly, apply glue to the second outline, spread very thin and evenly again, completely covering the outline and going past it some, and then carefully fold the fin over the leading edge onto the other outline, holding the fin and paper tightly so there aren't any wrinkles. Press down firmly. Using a Sharpie marker or something else round and smooth, burnish down the paper onto the fin and squeeze any excess glue out around the edges, starting from the center of the leading edge and working outwards toward the root, tip, and trailing edges. Go around the edges of the fin, burnishing the paper down to itself all the way around the fin, flip and repeat for the other side. THE most important thing is keeping a VERY VERY THIN layer of glue applied to the paper-- excess glue will only WEAKEN THE BOND and WEAKEN THE PAPER, creating wrinkles and tears in the paper as you burnish it down. Your finger should only require a swipe with a paper towel when you're done applying the glue to clean up--
Once the fins are dry (overnight), cut the excess paper back to within about 1/4 inch of the edges of the fin, then using a SHARP hobby knife shave the paper off the tip, root, and trailing edges of the fin down to the balsa... put a sheet of 220 grit sandpaper down on the work table, and holding the fin upright with it's tip, root, or trailing edge against the sandpaper, tilt the fin to a SLIGHT angle, and gently draw the fin towards you across the sandpaper a time or two, flip, and repeat-- this will remove any stray "paper hairs" that the hobby knife may have missed and leave you with perfectly dressed and finished paper edges, ready for the fins to be glued onto the rocket.
I posted pics of the process over in my Dr. Zooch Vanguard Eagle beta build thread...
https://www.rocketryforum.com/showthread.php?t=13600
Works like a champ! Fins are VERY VERY strong and ready for primer/paint, and it's permanent and cheap and easy to do... and having the paper continuous over the leading edge means no fraying/lifting label edges in a year or two... and no doping the edges with CA trying to keep everything locked down...
Later! OL JR