The RT 65B should be fine (and iiuc fairly robust against shorts etc, even if I haven’t tested that in practice 
), and the wiring in your photo looks nice and clean, so I don’t think the problem is on that side.
Your next step now is to unplug your module, optionally remove the ICs, and use your multimeter to make sure you have connectivity from the two 12 V rails to where you should have 12 V power, and no connectivity between the 12 V rails and ground (if you keep destroying the 10 ohm resistors, odds are the latter is your problem) (if you don’t have a multimeter, you need to fix that first).
Feel free to open a new topic and post some pictures of your boards if you want people help you look for potential problems.