Welcome to TE!
First off, what kind of bike is your Giant? (FCR/Dash or Rapid? Mountain bike?) And your drop-bar bike? Do you have difficulty steering? Are you physically uncomfortable on the bike? You do say you feel like you're balancing on the bars. Do you find yourself putting a lot of weight on your arms and hands? If this is the case, you may need a fitting and/or a new saddle.

This may not apply in your case, depending on your answers to the above questions, but: There's a part that's in your head, but it rapidly becomes physical. If you're intimidated by drop bars, you get nervous on that drop-bar bike. Road bikes are designed to be light and responsive, and that quality means that every nervous twitch or spasm gets magnified and results in squirrelly handling. No fun. The biggest thing you can do is make sure the bike fits properly, and relax!

Edit: The shoulder-width thing is a starting point, as are all rules-of-thumb of bike fit. Some people naturally hold their arms a bit wider than shoulder width, and if this is the case for you, the bars may well be too narrow.