Ive battled the dreaded "bit wobble" many times, I hate it. I do have some short bits, I even have a custom made 5/8" bit that is only 3 inches long, that I use in my metal lathe. It still wobbles on some woods. It works great on striaght tight grain like regular maple and such, but still wobbles on many others. And all my bits are sharp. Sharp bits and a slow entry technique are what is required but I still have to many wobbled holes.
Some places like Mcmaster Carr or Grizzley carry Deming bits, which are shorter than regular bits. You might start there.
The simplist alternative is to just drill on a drill press , and turn the striker heads on a blind mandrel or just between centers. Thats what I do.(blind mandrel)
Ive went from drilling on a press to switching all my drilling over to the metal lathe, then back to the press over the years.
Oh, also, I do have a 1 piece closed reed that I make, 5/8" barrel id. I drill a 3/8 or bigger , hole, then use the boring bar to open up to 5/8. This ensures I have a clean straight hole, that,in this case, is also flat bottomed.
Something to think about, although it takes a little more time.
Hope this helps.