British Airways, American Airlines, and Finnair have round-trip flights from Dallas (DFW) to Rome or Amsterdam (FCO, AMS) for $535-$584 with a connection. Flights October to March. Departures on Monday, Tuesday, Wednesday, and Sunday.
Amsterdam fare is more expensive than Rome.
A Carry-on bag and also a personal item is allowed for free on American. You can avoid paying for seat selection by accepting the randomly assigned seat at the gate.