Is Sintra a good place to stay for a 10 day trip?

Try airbnb it seems to have a number of places on or next to beach -- also in booking.com there is an option on left hand side to filter searches to 'beach' properties, plus if you select any property and click on it you will see the 'show map' option near top of screen that shows map of available places so you can work out what is close to beach

There are good, sandy beaches in Cascais and at a number of points east as far as Carcavelos. Use a mapping program (google maps or similar) to determine distance to the beach. We spent a month one summer in Sao Pedro do Estoril in an apartment about four blocks from the beach. Very pleasant.
With little kids, I would go with Cascais if you can afford it. Lots for children to do in Cascais, and there are good transportation connections to Lisbon and Sintra. Monte Estoril is another possibility, but a bit further from the beach. Other than Cascais, the best urban beach along that coast is at Carcavelos, the downside being that the beach is about 1 km from the commuter train station. Having said that, you might be able to find an apartment closer to the beach.
You can rent a car in this area, but parking is a big problem, and I suggest trying to make do with public transportation. Note that taxi's are easy to find in many areas, and are among the least expensive taxis in Europe. As noted by others, there are also good networks of commuter trains and buses.

We're having a hard time finding lodging. It's totally full in the Algarve and nearly so everywhere else. So I guess my essential question is this: does it make sense to travel all the way from the states with three little kids to stay in Cascais and not be able to stay in the Algarve? Aren't the beaches of the Algarve one of the major sites in Portugal? Are the beaches in Cascais nice enough to make the trip? Or should we wait until next year and book
Something farther in advance?

You have asked the same question 4 times - people have given their best opinions and links, at some point you have to make up your own mind.

Yes, true, I'm still struggling with this decision. And this forum always helps me sort through a difficult decision. Just looking for the one clear voice that will quiet my fear that if we skip the Algarve, we won't be missing essential Portugal. Sorry to ask so many times. Last minute travel planning is never easy, but is much worse w kids.

Self and GF have made four trips to Portugal totalling six weeks duration since 2013.

Although we have spent time on the Algarve - two weeks in three different towns over two visits - there is no way that I would describe it as essential Portugal.

Cascais would make more sense, especially with children.

Self and partner discussed this last week as we were talking about her heading off to Portugal with her daughter, son-in-law and her two grandchildren (2 and 4 y.o) for a break.

From Cascais you can head out to Lisbon, Sintra, Belem and the beach without a car as there's good public transport in the area. Others have posted thoughts on how good Cascais is for kids and on using public transport rather than a car.

You really need to make up your mind,

Carpe diem - sieze the day (or the pastella if you go to Belem!).

Carpe diem - sieze the day (or the pastella if you go to Belem!).

Especially if they're still warm, and topped with cinnamon.

Oh yes - with a bica, Sagres/Super Bock or one of each!

We have found a source of both pastellas and Sagres here in the UK.

Our local Co-Op store for the pastellas and either large Sainsbury's or large Morrisons stores....

Cinnamon stays out on the worktop, not in the cupboard...

In my opinion, Cascais is a good place to take your children. The beaches there are quite nice and you can have a look them using the google image search. I took my kids there when they were little, and have recently taken my grandchildren there. Many of the beaches in the Algarve are beautiful, but kids usually don't care that much for scenery. I find the most popular beach towns in the Algarve to be too touristy, and I much prefer the more mixed nature of Cascais and the residential/tourist areas to the west of Lisbon. There is much more for you to do in the area between Lisbon and Cascais than you could possibly do in 10 days.

Thanks again for everyone's advice. We are staying in estoril in a house we found on Airbnb.