Door County, WI.
Definitely not Napa, but go up the valley to St. Helena or Calistoga, and you definitely have your small tourist towns.
Hood River, OR is a really great little town, too.
I wouldn’t call it a treasure, but Tombstone AZ is definitely a touristy town. Pretty much all it is and covered in a few hours. The sunsets in southern AZ are gorgeous though, and if you drive up to Gleeson you can stop at the rattlesnake trailer. When the wind blows through all the junk they’ve collected, out in the middle of nowhere, it’s a lovely eery lonely sound.
Santa Barbara is like ten times too big for the OP’s requirements.
Oh, right, I didn’t notice the specification of 10K population-- Bozeman is somewhat over that.
That was sort of my point- most of the town (at least from what I’ve seen) is based around people stopping off for petrol, food, and a chance to stretch their legs. There was (at least IMHO) a pretty cool “bazaar” (for want of a better term) at the bus station there which seemed to do pretty well from transiting visitors, too.
I’m going to nominate Carmel-by-the-Sea (to give it its full name), California. It might be a little bigger than the proposed limit, but it is so precious I think it should be allowed in anyway.
There is little to do in the town itself except walk around and look at the pretty shops, or eat lunch. There is the beach, too.
Nearby is Pebble Beach, the 49-mile drive, Monterey and other interesting things.
Roddy
Ouray, CO, population 877, and Silverton, CO, population 531, except when the D&SNG trains are in town. 
Ouray is known as “The Switzerland of America” and is home to the Camp Bird Mine, which provided Evalyn Walsh McLean the cash to buy the Hope Diamond.
You can continue on to Durango, CO, which is definitely “touristy”, but too large for the list.
The OP called for populations less than 10k. There are over 75k people in Santa Barbara proper and over 150k in Santa Barbara/Goleta/Montecito.
Or Healdsburg, which is the cutest little town in the area IMHO.
A few nice little towns in my state:
~ Blowing Rock, NC
~ Ocracoke, NC (kinda hard to get to but well worth it, IMO)
~ Pinehurst, NC (only if you’re into golf, although there are some nice restaurants and such)
~ Bald Head Island, NC (like Ocracoke, it takes some effort to get there but it’s GORGEOUS)
Also, I don’t know if this would fit your population criterion, but one of my favorite small “towns” in North Carolina is Old Salem, a sort of small-town-within-a-city that comprises a pre-Revolutionary Moravian settlement that was later absorbed into the larger city now known as Winston-Salem. There are several restored buildings now open to the public for tours; the Old Salem district is reminiscent of Colonial Williamsburg but not as heavily commercialized, IMO.
Luckenbach, TX - home of country music festivals and concerts, but the “town” itself has a population of about 9.
And, of course, Radical Pi lives here so they knew that.
Seconded.
And for any fans of the TV show “Northern Exposure,” Roslyn, WA is worth a visit.
Santa Barbara is too big, but “weird little towns in the US that have a tourist economy” describes Solvang pretty well. It’s like you suddenly arrived in some weird Disneyland Denmark.
Other places outside of Santa Barbara are Buellton and Santa Ynez. I’m sure Buellton has less than 10,000 people, and it split pea soup that apparently lots of people go to. Santa Ynez is also tiny and has tons of wineries nearby, making good Chardonnay and Pinot Noir.
There’s also Solvang, the Danish town.
Yes! Door County, WI and Solvang, Ca.
Ojai, CA
Cave-in-Rock, IL
Bar Harbor, ME
Well, don’t forget the Shakespeare Festival. And the disgusting lithium water.
I did. What I did not know was the 10,000 population limit. My apologies.