I've never been particularly good at managing empire size, but I feel that several factors in New Civs is such that it's basically impossible to truly keep your empire size small.

First of all: Crew, the only truly viable way to get crew is to build the naval recruitment structures. which are planet limited to 1. With no way to reduce crew costs, and with several hundred needed per ship, you'll need a number of planets to maintain the ship upkeep. While officer jobs can get you crew, the buildings that give those jobs are also planet limited to 1. Pops also give crew, but that increases empire size and it doesn't seem to get enough crew on its own.

Second: Components and Dilithium and Duterium. While a late game Stellaris battleship might cost 1800 alloys, even mid game NC cruisers might cost 3k components, *and* a bunch of alloys, *and* Dilithium/Dueterium, which are primarily available only around other stars and certain planets. Components in particular need a lot of districts, which need planets and population. (fleet command cruisers are ~10k components)

Third: the sheer scale of the galaxy. the basic NC map is 1400 stars. There's a lot of room In each quadrant for some kind of alpha-predator faction. In the lower right, the Klingons *always* get super dangerous. On the lower left, the Tholians generally get quite dangerous, or the Breen or Cardassians. In the upper left, the dominon is *supposed* to get big and nasty. Of course, the Delta quadrant has the BORG. There's always going to be some bigger fish nearby. Because these factions are going to nab multiple homeworlds, they are going to have extremely powerful economies and fleets. On top of that, travel is always slow, so it's hard to properly doomstack your own forces. This requires multiple fleets, which needs fleet capacity and more rescources.

Fourth: lack of control over pop/empire traits/mechanics/origins. Unlike in basic stellaris where you can choose traits and civics/origins to your advantage, you just can't in NC. Worse, some factions have special mechanics which encourage or require expansion. The Klingons in particular. and the Federation has a big issue with size: as you add allies into the federation, you also outright take their space, which directly and drastically expands your empire size. (Sadly, you never get to take the homeworlds of those allies, which are usually the best planets.) Some factions even *start* with extra colonies and empire size already over 100.

You can certainly try to play tall, but empire size if this mod seems unusually difficult to control.