Hello,
The wallet.json is the important one, the attr files contains your settings. You will need to ask for a full resync on the new computer. Still, not advised to use them as backup files.
I hope the answer was helpful. If you have any other questions, we're happy to help!
Unfortunately, after receiving new funds into my GingerWallet and it showing as pending transaction, after the transaction was confirmed and I reopened the wallet, it didn't detect the receiving funds as if nothing was received into the wallet, even after a re-sync.
Need to resolve this issue regarding synchronization.