Travelling from Italy to Croatia

There are no inter-country public links like trains linking southern points in Croatia, so you'll need to get to Split first and use that as your launching point (unfortunately, I think many flights will go through Zagreb first). Once in Split, it's very easy to take a comfortable (public) bus down the coast to Dubrovnik. Stress free, inexpensive and great views. Split is worth a stopover regardless if you can spare 1-3 days.

Croatia Air flies nonstop from Dubrovnik to Rome. And it looks like low-cost carrier Volotea will start a Venice to Dubrovnik on May 28th, which might be too late for you.

Jadrolinija is a state-run company and Croatia’s national boat line, and as such the most reliable option when it comes to booking overnight trips between Italy and Croatia. The company also runs the most long-distance routes between the countries, with year-round service. The boat line's website, in English, is fairly easy to navigate and a good place to search for schedules and prices. Jadrolinija ferries connect Ancona with Zadar, Hvar Island and Split, and there are also routes between Bari and Dubrovnik from late March through late October. The bulk of the ships in the company’s fleet are large car ferries, so expect lots of families transporting their cars between the countries among your fellow passengers.

Trips depart Ancona in the evenings and arrive 7 hours later in Zadar or 10 hours later in Split

There is also ferry between Venice and Rovinj.
We went by ferry from Dubrovnik to Bari then by train to Rome. That was in October. You can do it the opposite way in May, I assume.

I would first look at flights at www.skyscanner.com; for instance, Vueling also flies direct from Rome to Dubrovnik for as low as $55. Then second choise would be the overnight Jadrolinija ferries from Ancona or Bari. Ferries leaving Venice run a shorter season and only to the north of Croatia, not Dubrovnik. Do you have return flights booked from a particular city?

The flights I am seeing are all over $300, and very long stopovers. Am I missing something?

Volotea flies nonstop from Venice to Dubrovnik on May 28th for 81 dollars. They only fly the route on Wednesdays and Saturdays.

Vueling flies Rome to Dubrovnik on Tues, Fri and Sat and then it looks like they even add a Sunday flight in June. Fares start at about 50 dollars.

Vueling has recently changed their flight schedule for the Rome-Dubrovnik route. I received an email this morning stating our late April departure flight was changed from a Tuesday morning to a Monday afternoon departure. Vueling is the most cost effective and efficient way I have found to get from Rome to Dubrovnik at that time of year. We selected the Optima fare so we wouldn't end up with expensive baggage surcharges at the gate. The flight for two ended up at 114 euro which included 10 euro in management charges and a 4 euro surcharge for using an AMEX card.
Several years ago we flew from Rome to Trieste in May on Alitalia and then caught a bus into Croatia (Rijeka) There we rented a car from Fleet and made our way down to Dubrovnik and back. We then caught a bus to Rovinj and spent two days there prior to catching the Venetian Line catamaran to Venice. It worked quite well. May offers very limited transportation routes between Italy and Croatia but with a little work, you can find a way to get there.