The question at hand: does dead man's switch proc twice in conjunction with Tel's ability.
Tel's ability is "the first time you would be destroyed, cancel any remaining damage, discard all damage cards, and deal 4 damage cards to this ship."
Dead man's switch reads "when you are destroyed, each ship at range 1 suffers 1 damage."
Now, to proc Tel's ability you must deal all the damage cards to the ship, rendering the hull at 0. "Destroyed" is defined as having damage cards greater than or equal to your hull value. So before Tel's ability procs, there is a moment where she is at 0 hull. This would define her as destroyed. Dead man's switch goes off. Tel's ability goes off. Dead man's switch goes off again the next time it's destroyed.
Thoughts?
The houndstooth title specifically designated "after" a ship is destroyed abilities may take place before a ship is removed from the table. It sets the precedent for this interaction to occur.