The solid blue light with the button pressed on mine lights for 8 seconds (it's the older type switch, before they tried to incorporate the "5-click" feature the new ones have). What that means is there's power going to the switch, and as far as can be indicated at that point, it should be getting from there to the atty or carto. My brain is too fuzzy to think right now so I don't know if you'd be able to take a voltage reading with a meter on that thing *without tripping the protection circuit*, which would shut off the power from the battery and give you no reading. (I'm experiencing a deficiency of either caffeine or sleep & in either case cannot access stuff I'm suppose to know.)
It is possible the protection circuits in the new switches flag for a "short" at a slightly higher resistance than the old ones did, but that's just a WAG on my part. Without knowing if / how much voltage there is off the switch-to-carto connector anything I say/write is wild speculation, though not without some basis in experience.
So, do you have a multimeter? If so I think I'd start by checking the voltages of charged batteries & (if possible) the voltage at the carto end of the switch (button engaged, light aglow).
If you have any known-good, higher res. cartos or attys I'd try one of them, just in case the protection circuit is being over-protective.
That's all I can think of for now, except... I'd suggest getting a spare switch first chance ya get. I keep one, and a spare charger, & a total of 4 batteries in rotation. Never needed the switch but the peace of mind has been worth a lot more than the $9 or so I paid for it.
Oh, one other question: did your kit include, or do you already have, any Noalox?
Best of luck!