I can't really comment on the specific places to stay on the North Shore, BUT, "walking distance to all things North Shore"? I doubt you can walk anywhere BUT the beach. The town of Haleiwa, for example, is more than 5 miles from Sunset Beach.

As for must-do's that don't cost a lot of money -- for sure Pearl Harbor and the Arizona Memorial, hike/walk up Diamond Head for a great view (if it's not a smoggy morning), drive across the island from Kailua to the North Shore on the H3 (spectacular), wander around Waikiki Beach.

I love Oahu but I always stay right in Waikiki, so probably can't be much help here.

Free and low cost things to do include going to the beach, swimming, surfing, snorkeling, taking The Bus around, finding free dance and music entertainment, Farmer's Markets, museums and historic sights.

I personally love the North Shore so I'd go with that. Sunset beach is fantastic. Is the house near Ted's Bakery? They have cheap eats and yummy desserts. Definitely get whole pies to share as it's much more cost effective than buying by the slice. But yes, you would def need a car if you stayed anywhere outside of Waikiki. Definite do's that are free/inexpensive - Pearl Harbor, Diamond Head, Koko head, Honauma Bay, KCC Farmer's Market on Saturdays, visit Haleiwa, all the beaches on the North shore, as well as Kailua and Lanikai Beach. Eat shave ice at Matsumoto or Aoki's in Haleiwa too!If you want to go into downtown/Waikiki area - Ala Moana beach park is great on weekedays and has free parking. Best thing to save money is to cook - pick up groceries at farmer's market or Foodland. If you want to eat out...try the various food/shrimp trucks on the north shore. Lot's of them scattered around. Loved loved loved Opal Thai in Haleiwa, but note that the place is very small and requires reservations, especially for that many adults.

Have fun!

Personally I would opt for the house close to Sunset Beach. The beaches in that area are really spectacular, and there is a walking/bike path for several miles making it easy to get from one beach area to another. I can't say I have ever been to the beach in Hau'ula, but I would imagine that it is not a really outstandingly beautiful beach.

That said, i would find out from the Sunset owner, or manager or whoever you are dealing with, the answers to several questions:
On which side of the highway is the house, beach side or mountain side. There can be a lot of traffic on the highway and if i would have to cross every time I wanted to walk to the beach, i would want to know up front.
Also, how close to the highway is the house? Traffic noise can be an issue, esp at night with windows open.
Exactly how far is the walk to the beach?
How close to the walking path is the house?

I would probably try to get each owner to email a photo of the beach in the backyard for Hau'ula house and the closest beach area for the Sunset house. Of course you may have already seen these on their rental listing or in your correspondence with them, but if not I would ask for it.