Wylder Hotel in Tilghman Island is extremely dog friendly and on the water.

Slightly over 2 hours away but Dewey and Rehoboth beach are very dog friendly.
Lots of vrbo’s and hotels allow dogs in the off season.

We have stayed at a couple different dog friendly airbnbs in Lost River, WV. And yes, Charlottesville has dog friendly hotel options.

OP, it's easy to find dog friendly places on AirBnb or VRBO.

We have stayed with pets on the Shenandoah river... Example: https://www.airbnb.com/rooms/19909471?source_impression_id=p3_1738849809_P3ShQFSmll39tP9Q

And a place at Saint Michaels on the water with its own pool. And at Lake Ana (we didn't go into the water but rented a boat for a few days, and went fishing etc).

Tons of options.

Not sure what you are looking for but this AirBnB near Lost River VA is dog friendly: https://www.airbnb.com/rooms/8440843

Boars Head Resort in Charlottesville is very dog friendly.

Lydia Mountakn cabins in Shenandoah.

BringFido
-that website

Quirk Hotels in Charlottesville and Richmond are fun and very dog friendly as are any of the Kimpton Hotels. We've traveled a lot with our dogs and it hasn't been too difficult to find places that welcome pets, AirBnbs and hotels. I usually just go to the Marriott/Hyatt/Hilton, etc website and filter for pet friendly. I do read the pet policy before booking as some have weight limits (which I think it stupid, but it is what it is).

https://www.redpointlighthouse.com/

We've stayed here with 2 dogs.

You can filter Airbnb and VRBO results for pet-friendly. I find that a lot out in the mountains are pet-friendly.

The four seasons is really dog friendly.

Looking for a 2 night getaway about 2 hours or less from DC that will welcome our new puppy. We are new to traveling with a pet so not familiar with hotels or rentals that are dog friendly and kid friendly. Would love any recs.