Keith, I know the effect you are noting. PVA has a bit of flexibility even once dry, which is essential with some papercraft models that are designed with less rigid structures - so the PVA joints allow some flex when handling. The long 'grab' time is due to applying the PVA while too 'wet'. You can try this by cutting some strips of a single paperweight and then applying glue of varying viscosity. To reduce the moisture content just let the glue 'rest' on the palette before applying. Depending on factors such as paper thickness, absorbancy, area being glued and initial water content of the glue, it will take different 'rest' times to get the optimal 'tack' in the glue. If you repeat the same 'tests' using thicker or more dense papers, you will get different behaviors. It should be apparent that it take experience to get the feel for this process. Probably, for any given paper model, it would be prudent to use some of the scrap to practice your gluing. I suppose this is why you can call paper modeling an art (among other reasons)! At least, with digital designs you print yourself, if you at first fail, you just print a new copy and try again (I once built four separate 'Tinkerbell' heads - not counting the attempts at the face alone- before I got one I liked.)
The challenge of building clean paper models is one reason I feel all modelers interested in expanding their skills should try it at least once (I mean, build one, though it may take several tries to get it done!) If nothing else, you can learn a new way of thinking about a physical model...
R./ Robert
