We will be arriving in Shannon two days prior to a 7-night stay at the Adare Manor in Adare. Our current plan is not to have a car rental for the first two days while staying in Shannon. Where can we stay in Shannon that is walking distance to pubs, restaurants and public transportation or would we really be better off to get our car a couple of days early?
You'd be better off to get your car early and stay in Lahinch or Doolin for the two days. There is really nothing in Shannon, and there's so much to see over toward the coast.
Definitely get a car. I think there are a total of two pubs/restaurants in Shannon and they're not within walking distance. You really can't walk to anything in Shannon, and really, there is nothing to walk to. You could rent a car and stay in a place like Ennis for two nights. There are a lot of walkable places in Ennis.
Shannon is not much more than the airport. Lots of people stay in nearby Bunratty, but there's really not much to do there either. The suggestions others have made for Doolin, Ennis, and Lahinch are good ones. Lahinch is my personal favorite.
Another vote for the car. There is much to see and explore. Adare itself is an interesting town. Get away from Shannon which (I think) is unattractive and a little congested so that you can enjoy beautiful Ireland.