Okay this is fun. So first of your amp reading is no Bueno. As I can tell you have no real load on the circuit so your amp draw will be low. Until something acts as a load like a light bulb contacting the frame and the neutral line or your hands. So the amp will rise.

Now the next thing there is power going to the motor, with out the foot pedal engaged?

A quick hunch makes me say is the new foot pedal wired backwards? On ac circuits it’s you cut power at the switch and the rest of the unit is connected to neutral line all the time.

Those sowing machine motors are typically “universal motors” they are brushed motors that can run on both ac and DC. So power flows from footpedals to brushes to armature, to windings back out the other brush. It’s a bit on the weird side that the shaft has voltage. It could be from a brush sitting funny maybe. But I would most definitely just check the wiring orientation for starters. Then go from there. Adding a ground connection isn’t a solution. A ground wire is to prevent you from getting hurt when this happens.

Well I would start of by testing the shit stained components on back bottom right. That looks to be the most significant amount of damage.

Or you can do it the good old read the smoke signals. And force power in to by putting a bigger fuse and letting the part that’s fucked burn up and show you the way. But you kina only have one chance at it.

What is the load on the relay?