How could I test for this?

You could measure the supply currents drawn with and without the sound working.

 

Is this running off of the chassis battery or do you have a deep cycle stand alone battery (house) to run this off of? Is it the only thing running when you are doing these tests on this battery?

Yes I have a separate deep cycle battery and the TV has been the only thing running

 

I would have the battery load tested and capacitance tested to see what state it is in. The TV may be another story but lets start with the power source and make sure it is good before we look elsewhere.

 

6 amps is a lot for a small camper TV our 18" screen samsung LED backlit TV uses about 2.5 amps at 12v including inverter losses.

 

I measured both at the battery terminals and at the wired sockets a and the voltage was the same.

Was that with the TV load?

When I measured the current to the TV the voltage was below 12v and so the TV would not power on with the multimeter attached.

So it sounds likes there's a voltage drop between the battery and the TV due to wire resistance.
Can you try connecting the TV directly to the battery?

 

You need more battery power or less TV. My 19 inch Element TV only uses 1.5 amps, but it will still work down to 11 volts.