Croatia flight plans

Best strategy is to plug your dates into Google Flights so you get the most accurate results.
I have gotten there from DC with one layover several times, so I imagine it is likely possible from NY. I'd choose a layover in Vienna or Munich over London, Amsterdam, Paris, Frankfurt if possible.
I would do multi city search into ZAG and out of DBV (and vice versa) if both are in your plans, will save you time.

My family did it last year on Finnair. It was the least cost and good service.

Zagreb is better serviced than Dubrovnik. But many itineraries involve flying into one
and leaving from the other due to the configuration of the country.

You can look at the websites for each airport and get some idea of what connections
they have.

I think just about every airline will have a 1 connection option into one of those
cities, but not necessarily every day. So you may need to shift by a day or two
when you get down to specific dates.

If you give an approximate month of travel, you may get more replies.

United was running direct flights from NY to Dubrovnik during the high season. They didn’t have daily flights, just 4x a week. Check with them.

I usually fly Lufthansa to Zagreb via Frankfurt. Lufthansa is a Global Alliance partner with Croatia Airlines and connects you and luggage ok if not too tight.

Flying back to Zagreb from Durovnik and reversing your route to the USA might be simpler than other options.
Lufthansa is still my favourite airline.