Since the only problem seems to be your old computer not syncing with the blockchain and you are using Bitcoin Core, you could try copying your wallet.dat file to another computer, download the same wallet there and sync with the Blockchain.

Then move all your coins to another wallet, even a lightweight one like Electrum where you don't need to download the whole blockchain to spend/receive your btc.

If you are using Windows, your wallet file can be found in the folder %APPDATA%\Bitcoin.

Some articles on how to backup and restore your wallet in a new pc:
https://en.bitcoin.it/wiki/Backingup_your_wallet
https://freedomnode.com/guides/14/how-to-backup-and-restore-bitcoin-wallet
https://bitcoin.stackexchange.com/questions/10974/import-wallet-dat-into-a-new-bitcoin-qt-client