Take the bus from Sher'm to SalThouse/Cley, and walk back along the beach to Sheringham. Admire the slumps & look for fossils along the way. 6-7 miles. Bring snacks & sarnies. Bus time table. Tide timetable here.
In other direction you can walk to Cromer and back (better on beach, need low tide), or even onto Overstrand and back. There's a nice clifftop stretch/walk from Cromer to Overstrand. And a fab cafe in Overstrand.
You can walk to these NT places, but quite a hike roundtrip (esp. for 7yo). So taxi instead to Felbrigg or Sheringham Country Park. I don't go on the Hall at Felbrigg, we lark around in the woods. Find the viewing platform at Sheringham CP.
Your 9yo & 15yo might enjoy Hilltop for activity stuff (climbing, swinging, shooting, ££). You could take the train to Hoveton & taxi over to Bewilderwood (£££).
If weather is bad then go to Splash in afternoon (shorter wait, may not be chucked out after only 1 hr). Still costs £25 for whole family.
It won't flop, it's a lovely part of the country for a holiday, and whether with or without a car there's a lot within easy reach.
Also, as you say outdoorsy, there's surfing on the east coast (certainly at Cromer and I think at Sheringham too). It's not the most exciting, but is something else to do on a beach, and might appeal to your eldest.
There's a lovely retro sweetie shop in Sheringham.