Test one controller at a time with every connector disconnected. Set the meter to measure resistance (omega symbol). Put your red probe on the red battery wire's pin and the black probe on any of the three main motor wires (like the ones that melted). Take the reading. Repeat for the other two motor wires. Put your red probe on the black battery wire and repeat the three tests. That will give you 6 results.
Each group of three should be the same as each other, but the two groups might be different. The results should be in the range 3k to 25k, though what's important is that they're the same as the other two in the group.
If the reading continuously climbs or goes down, try reversing your probes to stabilise it, otherwise just record what it shows instantly.
Do the same for each controller to get 12 results. This test should show if the melting caused any damage, then we can go on to test that everything is working.
This test is for the MOSFETs in the controller, which are like very fast high frequency electronic switches that control the power to the motor. If they blow, their resistance changes.