I believe you are...
A ship's captain. The lines with "my" may refer to either the ship or her captain, for they are synonymous to an extent.
I'll leave my slip behind you see
And the dress comes off when it's only me
As others have mentioned, a boat sets sail from a slipway. A ship's "dressing" refers to the flags they fly to relay information or messages. When a ship is by her lonesome or occupied/seen by none other than her captain, she has no need to communicate.
The proper fathom's set to my draft
Or else I'll find another craft
The "draft" of a ship is the part of the hull that dips below water. If the buoyancy of the ship is not properly calibrated, it will not be very effective (or safe) for travel at sea, and, as captain, your best bet is with another craft.
The men I know, they toe the line,
Or else they'll learn to drink the brine
Sailors must follow the rules (alternately, a possible origin of the phrase -- line up with feet aligned to the planks for inspection and/or lecture) or risk being thrown overboard.
If they do leave, I'll stay my ground
I trust my lass, to be well found.
When all others abandon ship, it is customary for the captain to go down with the ship. Ships are customarily referred to as female, hence "lass". Shipwrecks are often discovered quickly if they are near their last known position, or even years later upon exploration of the deep. ...Alternate interpretation just for fun -- if we are talking the ship, if ALL leave, she may stay afloat (ghost ship) and she may be recognized in part by her figurehead.
Clue reworked for the sake of completion re: check into "well found" again comment by OP:
A "well deck" is one that is found lower than what you'd normally identify the ship's profile by. As such, once the ship goes down, you may be likely to find it on its side, or upside down, with the well deck more exposed than normal.
Wrong again. Ah well. OP provided a definition:
well found: adj. (chiefly of a boat) well equipped and maintained.