On the lower right of your screen you should have buttons … “snap, grid. ortho, osnap, dyn, lwt, …”. “LWT” activates or de-activates “lineweights”. If you “right click” you will be offered a settings option. Click on that and there is the lineweight AND a box to check or uncheck called “display settings”. I’m unsure if this is the problem or not, but it is ONE place that it could be. Just de-activating that “LWT” should be enough.
Are you drawing lines or polylines?. The polylines may have a width set to them. If that is the case, all you need to do is activate your polyline command, click to begin your polyline, type “w” enter and it will display what your polyline width is. Change it to zero.