In addition to what I just described, when I doubled my opponent, and he turned it around by offering a DRAW, my clock/timer was ticking down. If I had let my timer count down to 0, I would have been declared the loser of the game. 

Update: I just played someone else, and tried the same trick. My Opponent doubled me while I was slightly behind (but not totally out of the game), and I tried to turn it around by hitting the OFFER DRAW key. I saw the OFFER DRAW appear for me, but my clock was still ticking down. My Opponent did not respond, and I still had 2 choices: RESIGN or ACCEPT DOUBLE. I accepted the double and ended up winning the game. 

See question 1 in my previous post. 

Question 2: Why was my clock ticking down when I offered a draw upon being asked to accept a double? 

When your opponent offers a draw, you have a third option to simply continue and ignore the offer, which is basically the same as rejecting the offer.

How could I continue when I was doubling my Opponent after he finished his turn (but before my turn began)? My dice never rolled, so How could I continue? Plus, MY CLOCK was counting down. 

You can continue when you are offered a draw, which was the original question.

Could you please spell it out for me? I don't understand how I could have continued, when MY CLOCK was ticking down. I was under the impression that if my clock ticks down to zero, I would lose the game by default. Are you saying that if I allowed my clock to tick down to zero, the game would have continued as if no draw were offered? 

Case #1:
1. you offer a double
2. opponent's clock is ticking, opponent offers a draw
3. opponent's clock is still ticking. you can either
i) accept the draw -> game finishes
ii) ignore the draw offer and wait
4. opponent will have to either accept the double OR give up ELSE opponent's clock will count down

Case #2:
1. you are being offered a double
2. you offer a draw, your clock will continue ticking
3. your opponent ignores your draw request (see Case #1 3.ii)
4. you will have to accept the double OR give up ELSE your clock will count down

OK... that seems to make sense to me now, but I have 1 question regarding Case Number 1 (in your example). I Doubled my opponent. He then turned it around and offered a draw. My clock was ticking down at the same time that his clock was ticking down. I had less time on my clock than my opponent did. It was still supposed to be my turn, but I could not roll my dice until the draw or double were answered. If I did nothing but wait, wouldn't I have been declared the loser because of my opponent's stunt? After all, my clock would have run out before his if I ignored his offer. 

When you double your opponent, it is his turn to accept or resign, so it is his clock that is ticking down (after the initial 15 sec move time is over), even if he offers a draw meanwhile.