The following attractions would seem to fit your criteria in Anaheim;
Disneyland
Disneyland Railroad
Main Street Vehicles
Disneyland Opera House - First 50 Magical Years
Main Street Cinema
Enchanted Tiki Room
Jungle Cruise
Tarzan's Treehouse
Pirates of the Caribbean (two drops instead of one, but you can do it!)
Haunted Mansion
Pirate's Lair on Tom Sawyer Island
Davy Crockett Explorer Canoes
The Many Adventures of Winnie The Pooh
Mark Twain Riverboat
Sailing Ship Columbia
Big Thunder Ranch Petting Farm
Pinnochio's Daring Journey
Snow White's Scary Adventures
Mr. Toad's Wild Ride
Peter Pan's Flight
Alice In Wonderland
Dumbo The Flying Elephant
StorybookLand Canal Boats
King Arthur's Carousel
Roger Rabbit's Car Toon Spin (don't spin your vehicle, just ride)
Meet Mickey's House
Finding Nemo Submarine Voyage
Disneyland Monorail
Autopia
Astro Orbiter
Innoventions Dream Home
Buzz Lightyear Astro Blasters
California Adventure
Monsters Inc. Mike & Sulley To The Rescue
MuppetVision
Hyperion Theatre Aladdin Show
Disney Animation
Playhouse Disney
Soarin' Over California
Tough To Be A Bug
Heimlich's Chew Chew Train
Flik's Flyers
Tuck & Roll's Drive 'Em Buggies
Golden Dreams
Sun Wheel Stationary Cabins
Midway Mania
Golden Zephyr
Redwood Creek Challenge Trail
I would suggest you avoid these otherwise mild rides in both parks;
Casey Jr. Circus Train (develops pretty good speed and has a mild roller coaster effect around curves)
Jumpin' Jellyfish (mild drop sensation while hanging 20 feet in the air)
Orange Stinger
Sun Wheel Swinging Cabins (deceptively wild ride when it gets going)
Francis' Lady Bug Boogie (if the Teacups are a problem, this one is worse)
Disneyland and DCA combined have nearly as many "rides" as the four WDW parks put together. Plus almost as much entertainment in two parks as the four parks combined. On a Saturday, Disneyland/DCA will put on three parades, Fantasmic!, fireworks, High School Musical Pep Rally, and several stage shows. Even when you remove all of the thrill rides, and some of the other risky rides, there is still plenty to do.
You won't be able to do it all in just one day, but by cutting out a dozen thrill rides you will get quite a bit done. Remember that Disneyland is open until Midnight every peak day, which includes Saturdays in September. Still, you'll need to plan this very well and use Fastpass when you can to get plenty of stuff in.
I suggest you focus on all of the rides that don't exist in WDW, but being careful not to miss Pirates of the Caribbean, Enchanted Tiki Room, and Haunted Mansion.