[KenWittlief]

I missed the part that Al caught, if you hold the relay contacts open the motor stops (like it should)

but since it runs with the handle unplugged there must be a short or mis-wire somewhere inside the welder, not in the handle itself.

start at the handle connector inside the welder, if you have an ohm meter that will make it easier. With the unit unplugged from the wall ohm across the handle tigger connector in the welder. It must be shorted (near zero ohms) if the motor is running.

if not, the input (coil) of the relay has been wired to something other than the trigger connector. Its possible the wire feed motor relay is chained off a gas solonide relay, and that one is stuck on?

so, either you have to find where that trigger connector is shorted inside the welder,

or you have to figure out where the relay is miswired and correct it.

hold on.... (its fun debugging a machine over the internet).....
you said the trigger does control the gas solonoid - its very unlikely they have two switches in the handle, one for the gas and one for the feeder.... so the trigger circuit is probabally ok

and the wire feed motor must be chained off the gas solonoid circuit - so its not the wires to the handle connector inside the welder, its the circuit that drives the feed motor relay that is always on.

There is a bright side to all this. Someone tried to fix this welder and screwed it up, that is why it was for sale, and Ill bet you got a better deal on it than if you bought it new.

so..... when you get it working you got an awesome deal ! :^)