There are a lot of options. One good one is regular old wood glue (aliphatic resin) such as Titebond. It doesn't stink and cleans up with water. I like the original stuff better than the newer waterproof versions. You could even use something like Elmers white glue. Then there are old school adhesives like Ambroid and Sigment which you can still get. Not sure how they do with stain though. Heck you can even still get hide glue from some woodworking places.