Running out of cards with "Loss of Card" in Sequence
I would say that you loose the game since you don't have any cards left and hence no longer are able to follow the rules.

The rules state that when it's your turn you should select a card from your hand and put it on the discard pile. It doesn't say that you are allowed to skip this and just take a card from the draw deck.


  ...each player selects a card of their choice from their hand and places it face up on a discard pile...


But I guess that a house rule allowing you to just draw a card is a good adjustment of the rules. All in favor of being able to finish playing the game.

SEQUENCE Rules