I assume you're talking about
this kind of PowerCON, which means an entertainment lighting setting? (For those who may not be familiar, it's common for stage lighting equipment these days to use PowerCON connectors, with in and out on each fixture, so that power can be daisy chained from one fixture to the next.)
Time delay relays are a thing. To support the full 15-20A rating of the connectors/wiring, though, you may need to use a contactor and control it with a time delay relay. It wouldn't be a
soft start, just a delayed one, but that really amounts to the same thing if you can stagger the starting of enough pieces of the load. A true soft start usually involves either active control of the PFC stage of the power supply (IE, buying a PSU with this feature built in) or using a thermistor in series with the power supply input.
But I would be most concerned with why there are sparks flying from your switches! Unless you're in some sort of campy frankenstein movie that shouldn't be happening!