10 day Italy trip - Venice/Rome or Venice/Florence/Rome

Go directly to Booking.com and plug in your dates and other parameters to see what is available.
Have the three of you taken a vote?

You have 9 nights
I love Venice but would probably recommend
Venice 2
Florence 3 or 4-the 4th night allows for 1 day trip to Tuscany town-Siena, SG or Pisa for example
Rome 3 or 4

Your flights are fine
Always better to fly IN to Venice than out of

As above-go to Booking.com to see what is available
Your it may be determined by that

We love Hotel Ai Mori d'Oriente in the Cannaregio area of Venice, on the quiet back canals. We never want to leave Venice.
Sorry about the family emergency, life happens, but you are really late to be booking hotels. I agree, hotel availability may shape your itinerary. If you find lodging you like , book asap.
Do you like smaller areas or cities? Venice and Florence are smaller, and Rome is hectic and bustling. Are you ready for that craziness? You are probably too late for Rome sites that require timed entry. What do you want to see in Rome?
You could always do Venice and Florence and less time in Rome,,to see if you like Rome enough to return.
You can have the Venice experience without going into a church or museum. Just walk, ride the vaporettos on the Grand Canal, and maybe out to Murano and/or Burano.
Enjoy your family trip, and safe travels!
Have a great trip .

May 30th, your departure day, shouldn't really count. Even with an early afternoon flight, there will be very little you can do before heading to the airpost. May 21, your arrival day, may be partially shot due to being tired, sleepy, jet lagged, and having to get to the hotel from the airport. That being the case, you really only have eight completely unemcumbered days. My suggestion is to limit yourself to only two cities, and the logical ones would be Venice and Rome. If you add Florence, you'll lose at least a half a day in transit from Venice, and then at least another half a day in transit to Rome. Just my two dollars worth -- inflation you know.

^^^^ I completely agree with TC. This would be a much more relaxed and less stressful vacation.

Sorry
I didn’t even think about tix to major venues in Rome being unavailable

If that’s the case-most likely is-you might consider more time in Florence and Venice, less time in Rome

Venice 3 or 4 (3 allows for trip out to Burano)
Florence 3 or 4 -2 minimum for Florence sights then 2 for day trips if you wanted-let’s you experience more of Tuscany

Rome 2 nights-allows for day and half of walk around-which is our favorite thing to do in Rome
It’s a living history museum
Walk past Colosseum, peek into Forum from Capitoline views, RS Heart of Walk-do both ways, one in evening, one in morning, etc
Have to be here anyway for your departure flight

I would probably just visit Venice and Rome in the time you have, because it will be more relaxing and possibly give you opportunity to explore some less-visited, less-mobbed sites, take time for relaxing lunches or coffees or aperitivi. More of a vacation, less of a forced march.