When I bought my aircraft my N number came up with the same response on FAA's website. It turned out that a signature was missing on one of the pink slip forms. FAA sent me a 30 (or was it 90?) days extention in the meantime.
Another good tip is to get the written out of the way as soon as possible. It will be an anchor until you're done. But once you clear this minor obstacle you'll have a great feeling of accomplishment and progress.
I would say try to cram as many lessons into a short period of time as possible (that you are comfortable with). Agree on the lesson plan with you instructor ahead of time to measure progress and stick to it. I realize that might be hard to get some good IR WX in NY during the winter..!
Good luck!