If it were me, I'd be debt free today, Monday, at latest. 100K-(53K+31.2K) = 15.8K would be my bank balance by this afternoon.
I am not sure why you would count 6K of your daughter's money as yours to use. Either it is her money, or yours, not both.
If you still want debt, which I think is silly, you might try to refi your home with a home equity loan. That loan would be in first position, at a fixed rate, and for a fixed term. Regions bank did some thing like this for a person with excellent credit and plenty of equity. They got a 7 year loan at 2.6% when the prevailing interest rate on a 15 year was about a point higher.
9% would be an absurdly high rate in the US by today's standards, so you should at the very least consider refinancing to fix that.
Paying off completely is of course another way to fix that.
Which approach makes more sense depends on what you plan to do with the money instead if you do refinance -- if you can put it in an investment which yieds a higher percentage than the loan costs you, than refinancing the mortgage could actually be a profitable decision.
You also need to consider what the bank will want to see in order to give you a mortgage on the new property. I have no idea whether they'd be happier that you had no prior mortgage, that you had the mortgage but also had the cash to pay it off, or if they wouldn't care.
Remember that you want to be able to make at least a 20% down payment on the new property to avoid the insurance fees. That requires that you have a reasonable amount of cash-equivalent on hand. And generally, "house rich but cash poor" is an awkward state to be in.
If you get a mortgage on the new property, you can almost surely get better than 9% today. I refinanced my house a year or so ago at 4%.
You didn't say how much the new house will cost or how much of your cash you're willing to put to it, but just to make up a number say the house costs $200,000 and you'll pay $100,000 with cash. So you could keep the $50,000 at 9% and get a new $100,000 at 4%, or you could pay off the old loan so you have $0 at 9% and $150,000 at 4%. Clearly plan B is significantly less interest -- 5% x $50,000 = $2,500 per year.
If you can afford to buy the new house without getting a loan at all, it gets more complicated. By not getting a loan, you avoid closing costs, typically several thousand dollars. Would the amount you save on closing costs be more than the difference in interest? I think probably not, but I'd check into it.
If paying off the old loan means that your down payment on the new place is now low enough that you have to pay mortgage insurance, you'd have to factor that in. I can't say without knowing the numbers, but I'd guess PMI would be less than the interest savings, but maybe not.
Oh, I assume that you're planning to keep the old house. If you sell it, this whole question becomes a moot point, as most mortgages have a due-on-sale clause.