Yucky, evil plasticiser!
Yes, it does essentially need to all come out before the stickiness is going to stop. Baby powder is good for soaking it up and making it more manageable. However you need to make sure you still wipe it down before and after application of baby powder as other wise it clumps together or will leave a surface residue of dried up baby powder. You need to repeat the process until the stickiness no longer comes back.
In addition, you can use rubbing alcohol for a similar treatment.
There is no set pattern as to how long or show short a period it will take for this to finish. Perhaps you could display them now and then just retreat occasionally?