Why can't I loosen the adhesive to remove the speaker covers?

Nothing you're doing wrong, they're just stubborn. I used a hair dryer on mine, the glue eventually softened up enough. New covers can be had for just a couple of dollars, so don't necessarily try too hard to preserve them (they'll likely not sit right again after removal anyway -- new covers come with adhesive attached, so they're perfectly flat, and just need to be applied).

Now, that said, there is a black raised lip around the outer edge. When I opened mine the first time, I didn't understand which side of that should come apart, and did some damage to it (nothing functionally wrong, but cosmetic, and it bugged me). The lip goes around the speaker covers, but no deeper, so to separate from the metal back, go around the outside of the lip, not the inside. This is probably obvious, and I was just being dim, but, just in case, keep an eye out, as it's incredibly easy to break off pieces of that ridge.

One more unsolicited tip I got off an unrelated repair video: the power button isn't attached to anything, and is obnoxious to try to replace once it falls out, but a small piece of tape over top of it keeps it in place, so it's no problem.

Good luck with your repair! :O)