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One of my concerns in this is the tax cost to your mom.80 acres at even the lower number of $7,200 is $576,000. The tax, if sold all at once would be quite high (I am using some assumed numbers to get a ballpark - $576,000 less basis $50,000 less exclusion for primary residence $250,000 = $276,000 x 28% tax rate = $77280). This is a high tax and reduces the amount available to pay for your mothers care.
I would seriously consider selling building lots, one or two at a time each year, to produce the money needed for her care and doing it at the auction prices. If you can do this the gain is spread over several years, more money is collected, lower tax bite proportionately, and you can save the house for last or sell it first to protect the exclusion depending on your mother's condition.
I see no logical point to selling it at a below auction price. Then the tax burden will come from that and even less will be left than if it sold at the higher auction price.