Nearly anything on James Island in your price range will be in a safe area (most of James Island is a safe area). Drive around the island with a map. The neighborhoods off of Harbor View Rd are generally nice, especially on the harbor side of the road. They're mostly older neighborhoods with established trees and larger lots. Same with the Harbor View end of Ft Johnson Rd.
Another area to drive around on James Isl would be the Riverland Terrace area off Maybank Highway. Much of Riverland Terrace will be out of your price range but not everything. It's an area that can be walkable to a nice area with the closest thing Charleston has to an Art Cinema as well as a few good restaurants and night spots. There is also a small Sunday farmers market that used to be right in Riverland Terrace but I think it's moving to a new location at 1575 Folly Rd starting the weekend you'll be here.
There's a lot to explore on James but that's a couple of starting areas to look at. You should also stop in at the East Coast Paddlesports Festival at the James Island County Park happening that weekend. $1 admission to walk around, $10 to try out any and all of the kayaks, canoes, paddleboards, etc, etc that will be there. It's a top festival if you're into paddling in any form:
East Coast Paddlesports & Outdoor Festival | Charleston County Parks and Recreation
The park itself is worth a visit if you're wanting to check out James Island too. It's the best county park I've ever been to.
Also that weekend is the Festival of Houses and Gardens downtown:
https://www.historiccharleston.org/E...d-Gardens.aspx
Great way to see Charleston's Historic homes!
For fun, try out a carriage ride downtown or rent a couple of bikes for a South Of Broad St self tour. Or you can do that tour on foot.
You could take a boat tour out to Ft Sumpter if you're willing to spend an afternoon on it. A shorter way to get a harbor boat ride would be a Harbor tour or even the Water Taxi. If the weather's nice those can be great trips.
Check out Angel Oak on John's Island if you want to see the grandest of the grand oaks around here.
You're going to wish you had more time...