We did the opposite: stayed at the Swan one night then transferred to the Boardwalk the next day.
We rented a car from National, then used it and dropped it off at the Dolphin after we filled it with gas (like, a couple dollars, but they sometimes have a small print rule that if you drive less than 75 miles you need proof that you've put gas in, it is a rule that didn't used to be in place).
The next morning, early, we checked out and just walked over to the Boardwalk with our luggage and left it with bell services until our room was ready (which was surprisingly ready in just a few minutes!).
It's a short walk, so I'd just walk over with your stuff.
A taxi makes sense back to the airport, if there are more than a couple people. There are plenty of cabs around. I've usually paid about $50. You could also try Uber I guess?