Welcome!!!
I'm Jeff, traveling with my DW Mina and DD11 and DD5. We did the Western Caribbean itinerary twice before and loved it. The only challenge at the time of year we are going is that in Grand Cayman you don't always make it to the island. They tender there and if the seas are too rough.... We made it once and not a second time. Here are some tips.
1. Key West - we are staying on the ship. We've been there before (before kids) and got off the ship last year. Just not a lot to do there with kids. But it's an easy get off and get back on. The aquarium is okay - small but a nice touch tank area.
2. Grand Cayman. We have actually gone to GC for a week long vacation before and stayed at the Westin. On our first Western Caribbean sailing, we rented a day room at the Westin and took advantage of the resort, pool, beach chairs, etc. It was nice but not necessary because 7 mile beach is so nice and the water so calm that the kids can splash about with no worries. This time we will just find a place where they rent chairs and go for it. I've read about a few places like that.
3. Cozumel. We LOVE Paradise Beach. It's a beach park with amenities galore and great food and cheap beer.

You can either just park yourself there and so long as you buy some food and drink there is no charge. Or you can buy a wristband that entitles you to all beach toys (rafts, water trampoline, water climbing wall, on ground trampoline attraction, etc. etc. We tried Chakannaab Park last year but even after paying admission and checking out the beach area, we got back in a cab and went straight to Paradise Beach. Often the best day of our Western sailing...
Re dinner. Most nights you will be just fine in cruise casual - khaki's and polo shirt or Hawaiian shirt. Only on formal night does it get iffy - but then you can just skip the dining room and go to Topsiders for a more casual meal. And that's only one night. The rest of the time is probably more casual than you are even imagining. Honestly we have seen everything - including jeans and white t-shirts on formal night. So you aren't going to get kicked out no matter what....it's just a matter of how comfortable you are.
Great to have you join us. I used to live in OC (Irvine).