Interestingly everything is intact. I was reading more about my multimeter (should've done that before I tried doing anything 
) and it looks like it uses an inductor in the clip to measure amperage (it only does AC). Narrowly missed that one 
. I'll be more careful with my equipment and high amperage electricity in the future.
Here are the results using it that way:
Single solenoid engaged:
All solenoids engaged (they didn't all sound so I expect it cut off the maximum amperage):
Thank you for all your help and advice, I've learned so much just in this thread!