Snap, we are going for a week in August too!
I had a week there last year too, not with DH but with a friend and her two boys. So we had 5 children between us. We were so lucky with the weather, when we got there I emptied every last thing out of my car and put the back seats up so we could all go out together in one car but in the end we didn't go anywhere!
Obviously there is the beach, and even if the weather is not brilliant it's a nice wide beach to go for a walk on. You can do crabbing too, and there is the abbey at the top of the hill (English Heritage I think).
We went on the steam train as far as Goathland, then walked a path called The Rail Trail. I took a backpack carrier for my youngest, who was two, but the two 5 year olds and the two 8 year olds managed fine. I guess that trip would depend how good at walking distances your younger one is. That was a really nice day out, we took a picnic to have half way along the walk.
Not sure about fish and chips, we went to the famous Magpie Cafe and it was good but I wasn't blown away by it!
Save up all your 2p pieces for the arcades! Our kids managed to get a few pieces of plastic tat prizes out of the 2p 'coin push' machines for a couple of £'s worth. I hate arcades, the noise drives me mad, but if it keeps the kids happy for half an hour I can cope with it (just!).
There are boat trips you can do too. Like I said, we spent a whole week in the town with 5 kids and they were not bored so it's a good place to go. Lots of Beacon Farm ice cream helped too though!
We're planning a couple of trips this time, we're going on 15th August but I'm sure we can't be lucky enough to have another completely dry week! There is stuff to do on the North York Moors, I think there is a forestry commission place, and you can also get to Scarborough, Staithes and Saltburn for a different seaside day. Scarborough has a Sealife Centre, but I would try and find vouchers as it's quite expensive.
Have a good holiday!