When Apple replaces my screen will it be water resistant?

It should still be somewhat. However, Apple's documents state that water resistance degrades with time. They do what they can when they replace the screen to ensure that it is good at the time. You can ask the technician that replaces the screen what Apple says at that point.

My understanding of how they're supposed to do it would be that they use a new adhesive seal each time. It's mentioned in the following article with a photo of one of these unused seals still attached to its backing as well as a picture of a press (with timer) used to secure the adhesives. I thought that when they put it on the "Horizon Machine" that it checks everything for integrity along with various checks of display calibration.

https://www.reuters.com/article/us-apple-repair-exclusive/exclusive-apple-makes-iphone-screen-fixes-easier-as-states-mull-repair-laws-idUSKBN18Y0BF

That was my understanding as well, however I wanted to let the OP know that it will still degrade over time, and if it won't pass, then recommend an out of warranty replacement of the phone. Such as a bend, dent or other defect in the frame that won't let the screen seat and seal correctly.