Yeah, 4br with basement brand new for $350k is tough. I saw one in Woodbridge but you already said you weren't a fan of that.
Around market village in Smyrna, there's individual homes built all the time as tear-downs, so look around. For instance, some are being built right now off of Fleming Street, and some David Weekley homes are coming right there as well but I don't know if they will be built in time for you.
Since they are being done house-by-house in that area, aside from David Weekley, I would first drive around areas between Market Village and Ridge Rd such as Forest Hills and Smyrna Heights or off Spring Street right outside of market village across the tracks and take note of any builders putting things up and contact them. I'd also talk to city council members and get connected to builders, and look at the agenda of city council meetings for the last few months to learn about local builders to get in touch with since there's rezonings being approved every meeting https://smyrnacity.legistar.com/Calendar.aspx
If you're just looking for new-ish, and not a warranty, you may want to consider looking at homes built in the last 10-15 years as well since there's a lot of them (Vinings Estates, Woodland Brooke, etc). You may be able to get a 4br and even after a few do-it-yourself updates like granite countertops still top out around $350k, especially if you have cash and can do a foreclosure.