Hi Andy and welcome to the forum.
I would agree with the previous comments. I have used many types of glues on cardboard in the past and of them all Woodworking PVA is probably the number one for me. (Resin W for example).
Avoid the really cheap craft PVA unless you know the quality. Some can be quite watery and cause wrinkling of thinner card or paper.
I find PVA takes too long to dry so...
... UHU, Bostik and generic equivalents are my favourites, in the solvent versions. Use them like contact adhesive, ie apply and leave for 5-10 mins (while you cut out another part) to go tacky and they stick instantly. I don't like the solvent free versions and the solvent ones are still available so I guess there is a real difference.The worst has to be normal UHU, not because it doesn't stick but because of the strings of glue it leaves
The trick I've found is to let the glue run out under gravity with only the slightest squeeze of the tube. This makes it much easier to control and avoid stringing. You'll still get some stringing but think about the direction you move the tube away from the workpiece and it's easy to avoid spoiling a visible part.