Can I turn a 12V 5A (60W) supply into a 5V ~10A (50W)?
Yes, it is possible to get 5V@10A using your existing 12V@6A Power Supply. The only thing needed is a 'good' Step-Down Converter (aka: Buck-Converter). 
Because, your existing supply can deliver up to 72W of power, with the minimum efficiency of around 70% of Buck converter, you can get 50W at the output. And as the required voltage is 5V and input available is 12V, you need 'Buck' type DC-DC Converter.

You can build it your self or buy ready-made. Just google them!