My 2 cents on "doodle" in this chantey:

Stan Hugill notes an "old style of singing" where a /d/ was inserted before an /l/ in a word (pg 273 of the 1994 edition of SHANTIES FROM THE SEVEN SEAS). He cites two sources, one which renders "haul" as "haudle" (more like /haudul/) and one that turns "rolling" into "rodling."
(I messed around with this idea on a rendition I recorded )

This seems to be what happened to "do" -- or rather, to "do let" = do-d-let. The extra syllable, /d/, fits nicely to the rhythm and change to a new pitch that it coincides with.

So, in my opinion, there can be no "do not," "do tha," etc., nor do we need to search for an etymology of a word "doodle." It is just "do," transformed by a singing style/ornament.

Gibb