There are some sounds that work pretty well:
Shipboard has mostly fairly relevant sounds, just turn off the “droid bleeps and whistles” which are very R2-D2.
Adding in Shipboard’s “hull resonance” and “environmental systems hum” would be good for nearly any of the below interior rooms.
Alien Cantina’s “crowd ambiance” with some of Engine Room’s “hull resonance” and maybe “warp core”, combined with perhaps some of the crowd sounds from Coffee House 2015, and maybe Music - Chill or Music - Smooth Jazz, could make a good Ten Forward.
Engine Room, of course, makes a great Engineering. You may want to turn off the “plasma bursts” on a more reliable Federation ship.
For Science or Medical, using Alien Laboratory with varying elements on or off (“high tech medical”, “monitors” or “lab peripherals” for Medical, or using “computers in the room”, “low drone” or “low pulses” for Science).
The set Starship has some great entering/exiting warp sounds, and some good Star Trek-like computer voices. A lot of the computer sounds here are a bit more Star Trek-y.
While there are no one-for-one phasers/photon torpedoes, the Starship Weapons (Starfinder) have several reasonably close ones to not just Federation weapons, but Klingon, Borg, Romulan, Cardassian, etc.
There are some very close phaser sounds in Starship Battle, in the various laser elements.
If you take some damage, the universal one-shots have explosions, shattering explosions, and things of that type. There are some great sounds for continual damage, like the “plasma bursts” from Engine Room and “neon lights” from Market Street 2076.
Spacedock will probably see less use, but it has a lot of sound useful for a Shuttle Bay. Or for Engineering when they’re repairing.
The Devourer set has a great “probe launch” one shot, for those occasions.
Regular doors are probably the best represented by the Entry Door Open/Close one shots in the Star Island set.
While there’s no precise “transporter” noise, the Energy Portal Open one-shot from Quasi-Organic Vessel is a pretty good one.
I hope that helps!