If it were me, I'd try wrapping it in Saran Wrap or any cling wrap. IF (big if there) the problem is that the tape is drying out, wrapping with a cling wrap will ensure there's minimal air around the tape during storage for anything volatile to evaporate into.

I too have found that old masking tape won't peel off the roll properly and won't stick well. It could be that you're buying an expensive tape that simply isn't moving off the store shelves quickly enough. To address that issue, many drop in at any electrical wholesaler and pay the suggested retail price for a roll of any electrical tape they stock.