Is it possible for a 6-player game of BSG + Exodus to only ever have one Cylon?
You are correct it is possible to have only one Cylon.

This situation can occur if one player is dealt two You Are A Cylon or Sympathizer cards, and the remaining Loyalty card with that name is the last undrawn Loyalty card.

Using the standard Execution rules for Exodus, if you were human, you would add a You Are Not a Cylon card to the Loyalty deck when selecting a new character. This won't change the possibility of having only a single Cylon though, because the Loyalty deck will still have one extra card. If you were a Cylon, you just get moved to the Resurrection Ship, keeping both your Loyalty cards, which also doesn't change the situation.

The only exception to this, is if you were playing with both Exodus, and Pegasus expansions. Step 2 is revised to hand off the unrevealed You Are a Cylon card to another player. So you would have two Cylons in the game. This is covered on page 22 regarding Handing Off Excess Loyalty Cards 


  Execution - The rules for execution are slightly revised for the Exodus
  expansion. Use the rules as they are described on page 7 of this rulebook, even when playing with the Pegasus expansion, with the following exception. During step 2, “Prove Loyalty,” a player with one or more “You Are a Cylon” Loyalty Cards reveals one and gives all his remaining facedown Loyalty Cards to a human player of his choice, as described on page 12 of the Pegasus rulebook.