What mah is describing won't be cured by a snubber, most likely. It sounds like the AC is coupling through the coil of the relay and back to the MCU. We'll probably need to see the schematic and board layout of that area to understand what's going on.
I use SSRs (Solid-State Relays) for cases like this 'cos the trigger is usually optocoupled to the triac, giving added separation between AC and your electronics. The Fotek brand that you can get cheap out of China gets iffy with 3.3V logic, and really needs a 5V drive signal even though the specs say it'll run down to 2.4V input threshold. Due to the high inductive load of a motor, you might still need a snubber on the AC side of an SSR, so keep those PDFs (posted above by MaxHeadroom78) handy.