Draft in a chimney is a combination of chimney diameter, height, temperature of the chimney and combustion air. The offset and the antenna are rarely the cause. The height of the chimney top above the roof can be part of the problem. Wind pressure against the roof can reduce of even reverse draft. A tight house will not let enough air into the house to permit a draft up the chimney. The damper stops air from going up the chimney, it does not help it go up. If the damper does not close off the chimney, it is not working to keep heat in the house when the fire is not lit.
With no fire in the chimney, take a smokey match or wick and see which way the smoke goes, up or down. If smoke comes out of the fireplace into the room, you are starting with a reverse draft as cold air comes down the chimney. Open a window nearby about 6 inches for about 10 minutes then do it again to see the effect. If air will not go up a chimney you are going to have smoke in the room.
With the window open, build a small fire and let it run for an hour to warm the chimney to see the operation. If this works OK, you may be limited in the size of the fire that you can build.
If the smoke never leaves the chimney, drop a large rock down the chimney to see if there is a glass pane blocking the air. Installers sometimes use this trick.