William, I would start by consulting a professional real estate agent about what comparable sales in Wharton are, and whether you meet the 15% threshold. I'm pretty familiar with Wharton and would be happy to help.
1) Since the home is on the market for less than it's appraised value, is this a short sale or is it undervalued for some other reason?
2) Is the house in question the same or similar to other homes in the area, who also have assessed values in the $328K range? If so, even if you do get an appeal, it would probably be unlikely that you would get the taxes lowered. (see the brochure Laura provided in the link)
To answer your question, if you purchase the house, the appeal is your responsibility, and as others have said, there are guidelines and timeframes to do this. Good luck.
Keller Williams Realty Atlantic Shore