You have a multi-wire branch circuit feeding these two appliances. This means that it's a single /3 cable, with two hots and a shared neutral. Since the two hots run together in the same cable, you are getting capacitive "crosstalk" between the two wires. This is very weak, and you can only see it because you're using a DVM. DVMs are very sensitive.
The bigger problem is the breakers
And the biggest risk is in phasing the multi-wire branch circuit incorrectly, which will overload the neutral wire. The surest way to get that right is to use a factory provided handle-tie, either built in to a 2-pole breaker, or made by the factory to tie two individual breakers. The handle-ties are keyed so they won't fit on the breaker in a mis-phase situation. 2-pole breakers are much easier to obtain. Note that you will need a 2-pole breaker if you ever want to install GFCI protection.
Second, it should not be possible to turn only half the circuit off.
That is precisely to protect you from getting nailed like you almost did. The 37 volts wouldn't have harmed you, but if you interrupted the neutral, that would've! Bigtime. Interrupting a neutral on a live circuit can kill you, that's why neutral wires have insulation.
So, when you use a 2-pole breaker or factory handle-ties to tie the two breakers together, that takes care of that also.
Again, the reason not to tie them with a nail is because a nail will let you tie, say, the two halves of a duplex breaker, which is something you should not do. That would cause precisely the overloaded-neutral scenario I mentioned.