It seems to happen almost as soon as the town initializes, so I'm not 100% sure how to get ahead of it. It makes custom careers kind of annoying if there's this rush to move in/transfer sims to fill them (I definitely don't need 6 garbage collectors)

Threshold question first: are you running the game with NRaas SP progression not enabled under General Options as per the recent discussion here? Or is it now actually enabled but not seeming to honor its own settings immediately on game load? It's easy to fix if that's the case.

It ought to be enabled now - though let me test with an entirely new world and get back to you on that - b/c I don't think SP settings carry over from world to world, do they? So I guess I just wanted to know what immigration/job transfer settings or whatever I need to change immediately upon creating a new world so I don't have anyone moving in/transferring jobs without permission

Okay then, a few things.

1 - The story progression checkbox in the built-in Game Options should be off (unchecked). This prevents EA's story progression from taking over temporarily when a game session starts up and the SP mod has not yet entirely loaded. Script Mods like SP take a few game clock minutes, not paused, to take effect fully.

2 - Using Overwatch's Export and Import commands, you can export your settings out of one saved game (if you have not already done so) and import them into another without having to worry about forgetting a few favored options between saved games. You might want to create a set just for new starting out, worlds so be sure to name the exports with something meaningful. What eventually works well in Bridgeport might not be so great in other worlds, for example.

3 - If you really want to block ALL resident sims from moving or changing jobs, temporarily or otherwise, then on City Hall or an in-game computer

NRaas > SP > Town Options > Household: Allow Move as Family > False
NRaas > SP > Town Options > Household: Allow Move Solo > False (this is the default anyway)
NRaas > SP > Town Options > Career: Allow Find Job > False

or, more forcefully on that last one, Town Options > Career: Allow Progression > False (you probably don't want this one on False at the Town level for regular play as it will impact your active sims as well unless you make an exception for an Active caste or at the individual Household or Sim Options levels)

The SP Immigration Gauge is set to 0 (no immigration) by default. And SP Immigration, when it's actually engaged, only moves at most 1 new family into town per cycle of the Lots Manager. It's not something that happens rampantly or all at once unless you engage the Rapid Immigration feature.

If you want to be even more forceful about certain residential lots remaining unpopulated, then yank out the refrigerators. Sims will never move into a lot on their own, under SP or otherwise, if there is no fridge present.

I've lost track of the exact thread where I learned this recently, but another thing you might find useful is that the Testing Version of NRaas Careers allows you to tune the minimum and maximum coworkers for careers.
I think that would help alleviate the latter issue you described, if you'd like to adjust some custom careers to only fill one or two positions.