Hi Nick,
Yes, you can certainly use it without the adhesive, keeping in mind if you drop it the screen will most likely pop off; you're already well aware of the waterproof aspect, but don't forget about dust, pocket lint, etc.
If you can pick up a full body phone cover locally on the cheap, that would be a good way to mitigate those issues until you can do the actual screen replacement and reseal it.
I've let my grandson use his iPhone X without adhesive temporarily while I waited for parts; the difference there is that iPhones have those Pentalobe mounting screws that do an excellent job of keeping the screen on even without adhesive.
But yeah, I personally would go ahead and use the phone without the glue if I was in your position, especially knowing it's only temporary.