I usually use a plastic sandwich box and keep it in the fridge. It stores OK on the side but I find it does not last quite as long.
What recipe are you using. I use a cooked one and find that the dough sometimes becomes sticky if it is not cooked enough. It feels OK the first time but not after that. The other problem I have had is not adding enough cream of tarter, after a week or two we have a really wet mess rather than playdough.