Long story short, Disney will send you replacements for any of your damaged Disney DVDs! It will cost you $6.95, but it’s much cheaper than buying a new one.
Disney will replace broken or otherwise damaged DVDs of its own movies without requiring consumers to purchase new copies of those discs, although the replacements are not quite free of charge.
As explained on Disney’s DVD Help Center page, owners of damaged Disney DVDs need only fill out and send in the Walt Disney Studios Distribution’s (WDSD) Blu-ray and DVD Disc Replacement Form along with the damaged disc and “a nominal charge of $6.95 for per DVD and $8.95 for per Blu-ray disc” to obtain replacements.
Here’s everything you need:
Disney’s Help Page
Snopes confirming its validity.