Tax assessment is irrelevant. That is like looking up the price of Google stock a year ago and trying to pay that now. My point is that assessments are only done once a year and they are based off of old data. The market sets prices for homes, and this changes quite frequently.
You should enlist the help of a Realtor to assist you in the process. They can provide a list of comparable recent sales for which to base your offer on. Obviously you want to get the best price you can on a home. But you also want to be fair so as to increase your chances of getting the home. In many areas now it has shifted to a seller's market. The truly good homes are in limited supply and multiple people are bidding on them so the prices are going up.
Use the expertise of your Realtor to develop a good game plan to get the house you want.
Best of luck!