If you have the reservation numbers, you may be able to link them yourself online. I'm not sure if there is a difference b/c of the Costco part.
Once your reservations are linked, you can do pretty much everything online in advance. This is relatively new, I believe, as I do not remember it from 2016/17 but was the case for our cruise this June: you can even take care of granting permissions for minors when you go through the online check-in process. You can designate which adults in the linked cabins are allowed to take the children off the ship, which linked cabin members 13yo and older can check the kids out of the club/lab, everything! I went to guest services to confirm it was all good to go (as we were taking my niece, who wasn't in our room, with us on an excursion without her parents). It was all set, literally nothing to do on-board. You e-sign and choose a password for making changes when you go through the process. It was all very easy. I did everything related to online check-in and activities booking for our 3 linked staterooms, simply b/c I'm the one most familiar with it all, and I do recommend having one person do everything. It simplifies activities bookings, really, and everyone in your parents' cabin has silver level booking abilities b/c of your child in there. You can also designate who is in charge of each person's room charges, and that includes designating the lead in your own cabin as in charge of your child's charges. While this may seem pointless if they don't have charging privileges - remember that gratuities, paid activities,
DCL excursions, and the club/lab magic band will all be charged to the child. This was mostly seamless for us, although one charge went to the wrong place for my niece (who was also in her grandparents' room for space reasons).
If your staterooms are linked, DCL ought to pick up that you want a table together, but if you want a table for just the 6 of you - you need to tell them that, so a call is definitely in order anyway. That's the only other issue I'm thinking of.
Any questions, let me know. I did all of this recently for our extended family group. Like I said, these are pretty recent changes (regarding the minors), within the last 18mo, so not everyone is aware of the new online way to do much of this. I'm sure you can still fill out papers onboard, but it was so easy to do it online that that would be a waste of your time.