So when you use the carriage release does the carriage moves back and forth as freely as it should? And the issue is just with the carriage not advacing with keystrokes and when using the spacebar? Sounds like you've got an escapement that needs cleaning. Coincidence? Or maybe you dislodged a lump of congealed oil from a rail that ended up in the escapement? Who knows.

There could be another reason too. I don't know how you cleaned the rails, but if you've moved the carriage to its absolutely extreme ends and the ball retainers in the rails were properly positioned by the last person who put them in, it could be that you've now got a binding ball retainer. It's happened to me more than once.

This is odd, but I was working on another unrelated Model O and the same problem occured - except much worse. I was typing some test sentences, not even working on the rails as I was fixing sticky keys, and the movement because sluggish and then stopped completely. This happened quickly after I had written several lines without issue. The carriage moves when I hold the carriage release and the carriage return (without holding carriage release) is only slightly heavier than usual, but on keystrokes there is no movement. I have to push the carriage to get it to advance, and even then it takes a fair bit of effort.

I don't know what happened, but I do know this one is severe enough that it won't work itself out like last time. Basically, here's what I'm wondering: Is there a way for me to access the escapement without removing the carriage? If not, how should I go about removing it? I'm thinking a bit of dirt or dust fell from the rails and is embedded in the escapement gear.

It sounds to me like something got caught in a rail, and then was dislodged.  I remember cleaning a carriage rail on a Smith-Corona (which has about the same rail structure) with a paper towel.  Part of the towel broke off and lodged between a bearing and the rail.  It stuck, needless to say.  Then a little later, I was sliding the carriage backward and forth, and the little wad of paper dislodged and everything was fine.

Royal escapements are VERY picky. Royal used basically the same escapement design for a long time. And if it gets any bit of gunk (Well for me....) they freeze up. 

Have you check for things grinding? I had a typewriter repair where the tab set was jammed above (Yes, ABOVE) the rack, and completely froze up everything.