"Applying a finish over it": Medium to thick CA is better used in this case. Do Not use paper towels to apply when making thick layers. Instead, use an applicator such as thin flat foam applicator or something like that. I take margarine tub lids and cut rectangles about 2" by 1". These will be flexible and the CA is added onto one end and applied in a manner similar to adding putty to wood with a putty knife, but with a very light touch. This puts it on smooth and builds up layers fairly quick. Run your lathe slow. Not sure how low yours will go, but no higher than 700-750 RPM.
McCool is having some moderately cold weather. CA doesn't cure fast below 70°, but it can. Make sure your blanks are very dry; apply the CA in a warmed environment and have your CA warmed as well. Cool weather between 60° and 70° can draw moisture in as it cures and cause a whitish fog/cloudiness on it. Below 60° increases the chances. I use a hair dryer when I do this but don't get it too close or it can cause it to bubble. All of these factors have to come together as one and that comes with experience. The thicker the CA coating takes 10 to 30 minutes or more for curing and can take a few hours, depending on the temperature. If you have a heated shop or room, that helps. Let the room heat up to about 70° for a good hour before doing this so that the blank and CA bottle are at the same temp. (Different temps cause heat interaction that draws in moisture from the ambient humidity and can easily cause cloudiness.
How is the McCool high school sports doing this Year? (I'm aging myself but if I remember correctly - McCool has had quite a few state championships. I could be wrong.)