I am an adventurous traveler and the only times I would choose the Caribbean over Hawaii is with babies/toddlers, when I am very old, disabled or if I have less than a week. For a 2+ week vacation with tweens, Hawaii hands down. My favorite Hawaiian island by far is Kauai. It has amazing beaches (Tunnels!!) and hikes: the Kalalau trail (you can do it partially, to the waterfall) and Waimea Canyon. LOVED the Awa'awapuhi trail: https://www.alltrails.com/hawaii/kauai
The food in Princeville is awesome. There is a new One hotel opening there in Feb, replacing the St. Regis. Stunning location and vistas.

If you must stay closer to the Northeast, then try Costa Rica. The Caribbean is so boring. Unless you are a diver. I am, but still, we find it rather boring.
Can you increase your budget? $10k is not enough for Hawaii (or the Caribbean for that matter).
How are your kids missing 2-3 weeks of school in Jan?

Another thing to consider is how strong of a swimmer and you and kids are. We love to snorkel but the water in Hawaii was much rougher than Caribbean. It would pull me out further and it was a little harder to get back. This was Maui and Oahu. I'm comparing to Bermuda, Cayman and Aruba. And the water is much warmer in the Caribbean in my experience.

Consider also that you can fly to Cayman in 3.5 hours nonstop and Aruba in a little over 4.

My favorite island in the Caribbean is Roatan off the coast of Honduras. Though, not sure if it would hold your family's attention for that long unless all got into diving.

I love Hawaii but it doesn’t get as hot and that can be good and bad. Sun makes the water and snorkeling better because it got a little chilly in the shade. I think volcanoes and big island are really good but I also LOVE oahu— it is such a cool island with Pearl Harbor and North Shore and Kualoa. So much to do and see. So if you want to relax and do beach a snorkel, go to a Caribbean island. So much of Hawaii requires getting out and about that it isn’t exactly relaxing. But I love it so much. The vibe is it’s own thing.

Anonymous wrote:

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Anonymous wrote:

Anonymous wrote:

Anonymous wrote:Native Hawaiians do not want white Americans to visit. They have bee very vocal about it.

How locals in the Caribbean treat white tourists is much worse

Where? I’ve never had this issue.

USVI, Bahamas, Jamaica and several other Antilles. I’ve experienced it first hand on and off resort and have an uncle that has lived down there for decades.

I’ve been to all those without issue.

I’m sure it’s very much YMMV. But I will say I have seen it much more than I had in Hawaii.

Anonymous wrote:

Anonymous wrote:

Anonymous wrote:

Anonymous wrote:Native Hawaiians do not want white Americans to visit. They have bee very vocal about it.

How locals in the Caribbean treat white tourists is much worse

Where? I’ve never had this issue.

USVI, Bahamas, Jamaica and several other Antilles. I’ve experienced it first hand on and off resort and have an uncle that has lived down there for decades.

I’ve been to all those without issue.

Anonymous wrote:

Anonymous wrote:

Anonymous wrote:Native Hawaiians do not want white Americans to visit. They have bee very vocal about it.

How locals in the Caribbean treat white tourists is much worse

Where? I’ve never had this issue.

USVI, Bahamas, Jamaica and several other Antilles. I’ve experienced it first hand on and off resort and have an uncle that has lived down there for decades.

Anonymous wrote:

Anonymous wrote:Native Hawaiians do not want white Americans to visit. They have bee very vocal about it.

How locals in the Caribbean treat white tourists is much worse

Where? I’ve never had this issue.

Anonymous wrote:Native Hawaiians do not want white Americans to visit. They have bee very vocal about it.

How locals in the Caribbean treat white tourists is much worse

Anonymous wrote:Native Hawaiians do not want white Americans to visit. They have bee very vocal about it.

All Hawaiians are Americans. You mean white mainlanders. There lies part of the problem...stupid tourists that treat them differently. We only encountered friendly people there because we know how to act right.

Much prefer Hawaii over Caribbean especially when you want to do more than lay on a beautiful beach, January/February is a great time of year for Hawaii (whale watching!) and 1-2 weeks is a fantastic time frame to work with. However, your budget is a concern given flight and food costs, so I’m not sure you can make it work.

Anonymous wrote:We have been to almost every Carribean Island + Hawaii. We just got back from Hawaii. Ugh-- flying time is terrible for kids. Most of the Caribbean is much nicer. Try St. Lucia-- gorgeous!

This is where we ended up too. Hawaii is nice to do once but we just don’t think it justifies the travel when the Caribbean is so nice.

Just plugged in 3 weeks in JAN in Aruba and found a 2 bedroom in Nord at lux condos by GG for $170/night. FWIW.

Everything else is much more costly.

I’d love to see specifics for a 2-3 week $10k trip for a family of 4 to Hawaii or the Caribbean.

Flights will easily be $4k. (Plus Uber or parking.)

How do you stretch $6k for 2-3 weeks on an island? There’s no way you can stay at a resort or even a budget hotel for that amount.

What about food?

You might be able to find an Airbnb/vrbo somewhere. It won’t be oceanfront.

Curious if someone can build this unicorn affordable island vacation for $10k!

Given your list of activities and 2-3 week timeline, I’d do Hawaii over Caribbean. A lot more to do and in a lot of the Caribbean, there’s a lot fewer options for independent activities or excursions. Stay in condos rather than resorts given your budget and also because condos are more convenient with a family - you can do laundry, cook some meals and just have more room to spread out and relax. If this is the first trip to Hawaii, I’d recommend splitting your time between the Big Island and Maui. Volcanoes national park in the big island is awesome and perfect for an active family. Maui has a good mix of activities and beaches tho it’s more developed as a consequence. Kauai is beautiful and I do love it for hiking and scenery but it rains a shit ton there and especially so in winter. The hiking there can be challenging for some kids and pretty tough (and sometimes dangerous/not permitted) during rains.