I did first encounter of Prestige Spire on my Hunter last night. Drops showed in my feed on the screen, including the Grind Underfoot emblem (that's the only one I specifically remember), but when I went to check my inventory there was nothing there. None of my inventory was full, but either way there's nothing at my postmaster either. This issue only happened in first encounter on my Hunter, drops from the secret chest and the end of the jumping puzzle were fine, as were all drops on my Warlock. EDIT: I'm dumb, the emblem is a variant (I wish this were more obvious when it drops, but whatever). I'm still missing any gear drops from that encounter though..
This may be an emblem variant find the appropriate raid emblem if you can click on it you will see you can apply.