There will still be formal nights, and even with freestyle, you are expected to dress for it. The good news is that the definition of formal seems to be changing. We went personal choice on Princess, and women were wearing everything from nice pants suits to little silk dresses to "Sunday best" to full evening gowns. Men were in nice suits with very few in tuxedos.
I was on the Norweigan Sky last July. LOVED freestyle dining! Great for days in port. While we did have formal nights, the other nights I wore either a business-style pantsuit or slacks, sweater & blazer. We were in Alaska - at times I would simply come in from port, change my jeans to slacks, leave my sweater on & throw on a blazer, & go to dinner. I was never underdressed.
Hi "b.", On days that we were going to be in port from AM 'til we returned for dinner - we wore jeans, long-sleeved shirts or sweaters then layered with pullover sweater or sweatshirt. We always took our waterproof jackets (lightweight type like Land's End carries). Days at sea, the guys wore khakis (or similar) pants & long-sleeved shirts, or sweaters. Gals wore slacks & sweaters, or blouse (not t-shirt). Casual, but nice enough to go to the dining room for lunch. For dinner, most of the time the guys wore something similar to what they would wear to the office - nice pants or dress slacks & button up shirts, or pullover sweaters. Gals wore pantsuits or dress slacks with sweaters, etc. Formal nights you can opt to go to an alternative dining spot if you'd rather not dress up. "Formal" does not necessarily mean gowns & sequins, though you will see that. I am a pants girl!! I have a beaded jacket with satin pants & a velvet jacket with matching pants that I usually take for formal nights. I travel a good bit, and personally (especially with cruises), I like to take a few basic items like black, navy & brown dress slacks, nice sweaters, 2 blazers, and 2 pantsuits. I can mix & match the slacks, sweaters & blazers. I can dress the outfit up with jewelry, a nice belt, heels, scarves, etc. As long as I look "pulled together", I have always felt appropriate. Alaska is my favorite destination - have a ball. P.S. Pack binoculars!!!