Thank you for your replies. Here's what we did.
We stayed at the Barcelona Airport Hotel the day we disembarked from our Transatlantic cruise. It's a nice hotel with a shuttle service to the airport that leaves every half hour. Since we were flying home out of Barcelona in the morning we booked our stay there for the night before our departure. We were in Valencia for two nights so we left our larger bags at the hotel (they charge a fee) and used our day bag/backpack for the trip. We thought the hotel prices were reasonable and used our Chase points to book the rooms.
We purchased tickets for a day or two in advance online for the ALSA bus departing from the airport to Valencia. The tickets were around $43 per person one way. The trip was about three hours and 30 minutes, and the bus stopped for refreshments a little more than halfway. We purchased return tickets, also online, the day we arrived in Valencia. The return trip was a bit different in that we were on a double-decker bus and it did not stop until it reached the airport. We then took the shuttle to the Barcelona Airport Hotel and flew out the next morning.
The buses were reasonably comfortable but I think the train would have been more comfortable. If you're over 6' tall or large-framed you might feel cramped. We did have to put our day bags in the luggage compartment. We opted to use the bus to avoid having to travel from the hotel to the train station. The buses were full so buying tickets in advance if you're set on a particular departure time is recommended. The drivers did not speak English but many of the other passengers did so we could ask questions. Other bus lines are serving that route, so if the ALSA is full, you can try the others.
Other thoughts. We considered flying from Valencia back to Seattle but found the connections to be either too short or way too long. Also, the fares was more expensive. Valencia is a delightful city and definitely worth a visit. We spent two nights there and wished we had stayed three.