Exactly what happened to me. We are empty nesters and downsizing. I had a signed Purchase Agreement at $248K and my house appraised for $213K. The County Assessor had the property at $244K. The appraisal challenges have to do with the comps. The appraisers have to use 2 comps in your neighborhood. They don't have to be comparable houses, just have to be in your neighborhood, and in our cases a different and lesser desirable school district. Given the current rules, I don't see how real estate can ever appreciate.
Tax assessed value has absolutely nothing to do with current market or appraisal values.