The advice to pack clothing that can be layered is standard for Alaska cruises. The reason is simple: weather in Alaska can vary widely, even on the same day, and it doesn't matter if it is June. As amazing as it may sound, the area where you'll spend most of the cruise is a rain forest - something you'd expect to hear about South America but not Alaska. Temperatures can vary from the 40s to the upper 60s in the day and at night. Rain showers can come and go, or you can have sunny warm days all week. You just can't know for sure, so pack a compact umbrella, a couple of sweaters, a light jacket/windbreaker, blue jeans. But also don't forget you just might have days where shorts are fine!
Booking tours independently is usually fine, as the tour operators know the ship schedules well and they can be cheaper than booking on the ship. If this is your first time, and you are going solo, it might be a better idea if you stick with tours sold by the ship though. The peace of mind can be worth the extra cost. Ship sponsored excursions are typically more expensive (which is why people say book private tours), but they've rarely seemed significantly more expensive to me. Remember that most people are booking tours for 2 or more people in their cabin, so they are looking at a larger total cost.
Royal Caribbean's drink package for the non-alcoholic drinks doesn't seem like a good value to me, and it has restrictions on what you can get. Regular coffee and tea is included in your cruise price and is always available. I suggest you skip the package and just buy specialty coffees, bottled water, etc., by the drink. As you said - you are ashore a lot on an Alaska cruise. The ship is primarily your hotel and overnight transportation to the next port.
I love Alaska cruises. They are unlike any other cruise. Totally different character to them. Serene. Breathtaking at times, especially on the glacier days, heading up a fjord with soaring mountains all around.