How are you getting the private key(s) from your cold storage wallet exactly?  Are you using the dumpprivkey command?  If it's in hex format, you need to base58 encode but it's strange if you're using Litecoin core, the dump should yield a WIF private key.

edit: You better make sure you understand the Exodus wallet.  It's a one way trip for your coins if you don't!  You're not actually importing a private key, you're moving the coins to an address within the Exodus wallet.