For the mountains: Is your sweater cotton or wool? If it's wool, you'll probably be fine and can layer if you feel cold. A lightweight hooded rain jacket or one with a zip-out lining will help too.
I live in Florida too and am always bundled up on trips to cooler climates. I have learned that layering a camisole, a cotton shirt and a cashmere cardigan is often enough during the day and add a jacket at night or early morning. I bring a couple of scarves for my neck, even cotton helps keep my neck warm.
This may sound like overkill, but I blithely went to Ireland one May with a raincoat and a cotton sweater and froze. The first day there I had to go out and buy a wool cardigan.
Your trip is getting close, have a wonderful time.