Bestchange.com does not provide any service for exchanging, It is just a website that provides users with a huge list of exchanges for getting the best conversion rates. This helps the users to save a lot of time without the need for the users to compare rates on each site manually.

There should be an option for you to input your Litecoin refund address when you are using shapeshift for the exchange service, if you did remember to input the refund address, shapeshift would refund your Litecoin back to that address you provided should the system fail to do the exchange. And you should contact shapeshift instead of bestchange as your Litecoin is probably with shapeshift now.