I am guessing that you are running out of battery power. What I suspect is happening is you have a loose connection from the converter to the battery. I found mine in the negative terminal at the converter.
When you unplug the connection cools, then when you plug in a little power gets through to the battery, but then the bad connection heats up and sputters and the converter puts out full power, but the loose connection is just like a load, and takes all the power, leaving nothing to charge the battery or power the internals.
Check with your meter at the converter to see what the voltage is on the converter output, then at the battery. They should be almost the same. If they are not, the power is not getting through, either the negative or the positive lead to the battery, and the battery level keeps dropping.