@Actnaturally DH and I have done that crossing, and have fond memories. Well, actually we went from Fukuoka to Busan on the now defunct catamaran ferry and returned Busan to Osaka on the Panstar Dream, but I hope I can help. We travelled in Sept '23 and it looks like the ship was replaced by Panstar Miracle early in 2025, so my info may well be out of date. There is definitely a lack of info about this journey on the internet, when we went I could I only find one blogger's account.
As far as I remember it sells itself as a bit of a 'mini cruise' but we felt the Dream was more of a 'nice ferry' than a cruise ship, but the morning of our second day was spent travelling up the Seto Inland Sea under a warm, clear, blue sky which was lovely. It looks like the Miracle is modern and much more 'cruise like' as the Dream only had a buffet for dinner and breakfast, a tiny shop, and the onsen. I'll add a photo of our room, and we had shared bathrooms (for alternatives, the price catapulted and, frankly, we're too tight!). I would say it's much more interesting than flying...
I have just had a look back through my booking info and we booked through Direct Ferries so I probably struggled with the Panstar website. I have also just tried a dummy booking and it allows you to enter the children's ages but it does look like they are classed as adults. If you select a 4 person cabin they must be really unlikely to split your booking, so I am sure you will be fine with you all four together in one cabin. Did you find the cabin info on the fares and cabin info Panstar pages?
I can't help you with departures from Osaka as obviously we arrived into there. However, we have arrived into Busan port from our Fukuoka catamaran. I don't remember much so I think they must have been quite swift, and it's more like an airport than the ferry terminal at Poole!
Will you have eSims? If not you will need to get them at the port terminal in Busan before you leave the building and set off. We once made the mistake of not picking a sim up on arrival and it was a wild goose chase taking 3 days (I think) before we got the right info and right directions in Seoul to get one, so this is a STRONG recommendation even if you are at the back of a queue. eSims are SO much better, yay for a new phone! On our last visit we flew to Tokyo for a few days and from there flew on to Seoul for a week (DS lives there) so I bought an eSIm which covered both countries - easy peasy... DH just hotspots off me.
When you arrive in Korea you should also pick up T-Money cards from a convenience store (CU/GS25 etc), and you will need to load them as they come empty. I cant remember if you can pay by card in the convenience stores so I'll ask DH. We were there in Dec and I think the machine in the subway only took cash, but again I will check to see if DH can remember.
You can walk on a covered, high level walkway from the port to the train station in Busan. The bus station is about an hour north on the tube and is not very convenient.
Are you staying in Busan or heading straight to Seoul? When do you go?