From info sessions and other workshops I attended, the (typical) process described looked more like: yes, your app probably gets *read* in the order it's submitted, but after that (many) schools will "track" your app based on stats + the rest of your file.
(Barring ED/priority review/other options where the school would need to jump you ahead in the queue, or rather put you in a separate queue that exists ahead of the general RD inbox)
If you hit all the stats they need + your softs are great, you might end up on the "probable admit" trackand have a decision pretty quickly (whether it gets shared right away or not). Ditto if your stats are disqualifying or you come across as a serial killer in your app. But a lot of marginal applicants may get shunted to some kind of purgatory where their file gets a second look later in the cycle, or gets moved from "marginal admit" to "marginal waitlist" or vice versa, and the decision on your file is made after people who applied months later but were a clear call
The official line is "if you give yourself the time for Princeton Law School to take that 2nd/3rd look at your file, that may help (compared to applying in February so they need to make a snap judgment) but unlikely they fill their class by the time your file is read or anything that drastic
@ChromaticStorm: I know these are anxious times and there are a lot of unknowns, but believe in yourself and it WILL work out. 2 years ago I thought I would never get in and I'm somehow already a 2L. I believe in you