I'm still working this out, but...
I believe the key is to "Post draft to Weblog" (not "Post to Weblog" or "Save local draft").
Further, there is definitely some hinkiness relating to UCT and TZ offset.
If, at 7/3 3:20 PM HST (== 7/4 1:20 UCT), I set the publish date to "7/4 1:59 PM" the post is immediately displayed with a timestamp of 7/4 1:59 PM (i.e., it appears prematurely). If I set the publish date to "7/4 2:00 PM" (n.b. == "7/5 00:00" UTC) it does not post immediately. In other words, I think that "today's date" is based on UTC, but time is corrected for UTC.