Just to give an update, I did save the piece! I bought one of the Preval sprayers, and while I do really like it, it wasn't the best tool for fixing my botched work. I ended up using a sponge, applying generous amounts, and rubbing in circular motions as was suggested.
I will definitely use the Preval sprayer in the future. I think it will help me get a nice even coat the first time around.