Do you need to exit and check in again for a second domestic flight at Don Mueang if you have no checked baggage?

On domestic flights there is not much of an 'air side'. This term normally refers to being through immigrations on the way to international flights (or before immigrations, with arrivals). Domestic you still have a security check, but that is pretty much it. It would highly surprise me if they have a check-in desk on the other side of that, considering they also don't have this for international flights. But as others have said, if you check in online, you might slice out some of the time involved. Security should normally be pretty fast.

@Don ********

same airline?

There is a transit desk for each airline will a small security checkpoint as well.

Works fine for the same airline. If different airlines and checked bags it might change things.

Check in online and you don’t need again