getting to croatia

First, if you look at sites like Google Flights or Kayak ( there are others as well), I doubt you will find a direct (non stop) flight to Dubrovnik from Seattle. But there are several flights with a single connection either on the US east coast, or in Western Europe. I'm not sure why you singled out Paris as the city you would want to connect in, unless you have an unnatural affection for CDG.

Go to one of the above sites and see which of the available flights work for you. As well as price, look closely at the connection time in the connecting airport. I mean do you really want to cool your heels in JFK or FRA for 12 hours? Booking connecting flights on a single ticket has many advantages over cobbling together separate flights on separate tickets; not the least of which is that if your first flight is so delayed that you miss the connection, the airline will put you on the next available flight. Separate tickets? You'd be SOL.

Once you find the flights you want, book directly with the airline, not a third party. Its much easier if something should go wrong - you won't have to deal with a middle man.

Is flying the whole way the only option? Is Croatia your one destination? If you were planning on having a rental car anyway . . .

For what it’s worth, on our trip, we flew to Venice, Italy, and picked up a rental car. We drove east, then south, through Slovenia before reaching Croatia a few days later. We returned the car in Venice, where we stayed before flying home. Maybe extra countries and/or additional time aren’t anything you’d consider, but Venice was a good option for us, and we wanted to see some of Slovenia, too. British Airways flies Seattle to London, then connecting flight to Venice. Not certain how that affects overall affordability, but it might be an option?

IF you are travelling this year I'd be looking at direct options if they exist - even if they cost more. The fewer countries you transit through the less likely that borders will be closed and affect your trip

Maybe not convenient for you, but I chose a United flight from Newark direct to Dubrovnic because I didn’t want to change in Europe. I will fly from Dallas for my connection in Newark. However their last direct return flight is Oct. 3.
*Edit Fares were very reasonable a month ago.

This is what we found convenient from SeaTac in 2018. I can’t speak to affordability. We flew Delta to Venice (one stop in AMS). We spent a week in Venice and then took the GoOpti shuttle to Ljubljana, Slovenia. We started our Adriatic tour there. If you are just visiting Croatia, GoOpti does transfers to Zagreb, Croatia from Venice. Our return flight started in Dubrovnik>Zagreb>AMS>Seattle. GoOpti.com

Americans cannot currently transit AMS unless they fall in one of the categories of people who can be admitted to the Netherlands. It's really important to check very carefully before choosing a flight that requires a connection in Europe.

As noted, there are Covid related restrictions across Europe that keep changing. Be aware and pre-tested.
In normal times, I fly Lufthansa direct from Vancouver to Frankfurt, change planes to Croatia Airlines or Lufthansa and on to Zagreb.
This year the usual ferry is not running from Venice to Croatia, my other preferred route.
By bundling the flight with a partner airline you avoid airport changes and get a reasonable deal. Maybe not this year...
The French just blocked the Brits from coming in due to the India variant, the situation is still in flux.
Depending on when you go, the flights by smaller airlines may have increased. But you might get caught at a border.
Croatia has excellent bus and even train service from Zagreb to the coast. August and September look good.
There may be pent up demand that sees more airlines fly from the East Coast direct to Croatia as noted, they want US tourists.
Full details are on Total Croatia News which keeps tabs on all new flight options as they are announced.