Been there also.
Many wonder whats the mystic of the trumpet.
Its how to drill the right size, right length and of course a STRAIGHT long hole.
If that was easy trumpets would be a common turning
Yes I drill first on the lathe. Then it goes on a mandrel and away I go.
Keep trying you will find what works for ya.
1 thing that I found to help was to drill large and then step down. I tried running the small bit through and increase the size but I failed alot. If you got a straight small bore no problems but I found that the grain of the wood caused my small bit to directional bore even on the lathe. Amazing the flex they have.