We tackled the weather strip project today. Used the adhesive that Thomas G recommended, which we found at the local auto parts store. I think it will work beautifully! Instead of squeezing directly out of the tube, we put some into a small jar cap and used a small artist's paint
brush (which was sacrificed) to apply it. Seemed more manageable that way. The stuff is really stringy. The weather stripping was still in really good shape, so we didn't have need to replace the whole thing. Thank you, everyone!