I've stayed at the Swan and Dolphin a few times (and loved it), but they are different hotels really (I prefer the Swan).
Some trips, I've rented a car one way from the airport and dropped off at the Dolphin (Alamo and National do this - same company really - but be sure to "fill up" the tank even if it's a dollar, as otherwise they have a small fee if you don't provide proof that you filled up the tank if you drive under 75 miles). When I've done this, it was under $30 or so, and we would stop on the way to pick up a few things at Target like soft drinks, water, and snacks for the room (or wine). I've also used a town car service (there are several) with a grocery stop. Runs about $100+ round trip. Taxi runs about $50+ each way, but you can always take Uber back to the airport much cheaper.
It comes down to what you want: do you want to avoid driving? I'd to town car or taxi from the airport. Town car will meet you in baggage claim and assist with your luggage. Driving may be a bit cheaper, but then, well, you have to drive (ugh) and go pick up the car as well, but you would have the car all day if you want. Uber that picks up at the airport is the more costly one, and it costs more than a taxi so not worth it, but going back to the airport it's cheaper!
I just noticed it's your honeymoon - congratulations! I'd go with a town car! Make it special! You can schedule one with a stop if you want to pick things up at a store on the way!
