I just got off the phone with someone at the Recording Academy. Here are their preferences for how to render the word GRAMMY®:
GRAMMY®s (note the lowercase s)
GRAMMY® Award winner
So, the word GRAMMY is always all uppercase.
Source: The Recording Academy marketing department. Thanks for that!
Note: In running prose, common practice is to include the ® symbol in the word’s first usage, but after that, it can be omitted.