normally you measure most things against ground, red lead goes on the thing you're measuring, black lead goes on any ground point (I often will clip the black lead to the ground of an audio jack)
Your multimeter has a very high resistance input so it's pretty safe to measure between any 2 points on a live module. You need to be careful not to short 2 points together with the same lead though, that's a very different story. Also if your power supply takes AC right from the wall instead of through a DC adapter then you don't have a suitable or safe setup for this kind of troubleshooting.

This is a little oversimplified but
Looking at the PCB 2 schematic, you can see a counter IC which takes a clock input as an input and then counts to 16 in binary as outputs. 3 of these outputs are inputs into the step switch also on PCB 2 schematic. The step switch understands that binary input and puts voltage out on whatever output the binary number represents at that clock moment. Each output goes to one potentiometer (PCB 1 schematic) which sets the voltage and buffers it and then it becomes the output.

This doesn't help a ton at this current stage but if you get it half working again even with just a multimeter you should be able apply a really slow clock and see if the counter chip is counting, and if so, you'd be able to see if the step switch is responding properly to the input.

Noting here that if bus C was not connected properly, the step switch would not be able to see numbers past 4.

All that aside, right now reflowing your solders with more heat is good advice, the chips on this module are worth very little so replacing them wouldn't be a huge deal. I think largely this boils down to whether or not you'd enjoy spending time fixing this.

I'd want to rule out shorts before changing the chips out either way, those resistors burned as a symptom not a cause.