Apparently the tune, and the name The Sick Tune
relate to the words:
"It befell at Martin Mass,
When the weather waxed cold,
Captain Carr said to his men,
We must go and take a hold.
Sick, Ah Sick and Very Sick,
And Sick and Like to Die,
The sickest night that I abode,
Good Lord, Have Mercy on me."
from the Child ballad Captain Carr
(No. 178 in 'English and Scottish Popular Ballads')
As discussed here, viewtopic.php?f=48&t=56899#p621778
, neither the tune, not its setting have any connection to Dowland.
Nicely played, though; you give it a real melancholy sound; there s a temptation to play it faster, when it can sound unduly cheerful!