You as a home owner can use the proceeds as you see fit.

BUT the point is...do you think you or the agent will sell it for the highest price and provide higher net proceeds to your family member? 

Any advice on how to proceed would be greatly appreciated as I'm new to this. 

You will probably win when she doesn't show up, and while you will have a hard time getting the proceeds of the judgement, you can mess up a credit score pretty well with an unclaimed judgement.

Weirdly, you may actually be able to get better terms and/or more LTV on the refinance if there is debt on the property as lenders have rules for max cash in hand (max proceeds you get at closing) for cash-out refinances.  

Also the strategy you are implementing plays a huge part with how to proceed. 

Actually, it is possible do structure a 1031 Exchange on the sale of one or more relinquished properties and then using the proceeds through the 1031 Exchange to construct improvements on real property you already own.

If I refinance for $279,000 and use $70,000 of the proceeds to pay off the HELOC, will the interest on the full $279,000 refinanced loan remain fully tax-deductible?

Also, for anyone who considered it but chose not to proceed, what made you decide against it?

Any updates on how to proceed with this ridiculous law?