This is a pretty quiet part of the state, especially this time of year. Norwich has a lot of rental properties and is close to the casinos if that's your cup of tea. It's about 15-20 minutes north of NL. These are huge casino resorts so keep in mind that there's a lot to do besides gambling, like dining, going to the arcade, seeing live events and concerts etc. Waterford itself is beautiful but very boring. If you live in Groton, I'm sure you'll find yourself crossing the bridge often to go into NL. I just moved to New London honestly because I love the place and find it the livliest out of the towns you mentioned especially if you're in you're in your 20s. Great food, 3 colleges, live music, lots of cultural events, museums, indoor skate park, bars , and local art scene. It's also a bikable small city, and you can hop on a ferry whenever you want to go explore. It has dingy parts, but also really beautiful gilded age homes. Idk how you're looking for an apartment, but if you're looking for roommates, there are always postings on craigslist, that's how I found my place. Good luck with the move, and welcome.

Edit: just read that you are outdoorsy. No matter what town you settle in you'll be surrounded by nature.