I would like to pay off the existing loans and have a new loan (with the corporation as the debtor -- I understand I'll be giving a personal guarantee) for the purposes of having cash to use for a down payment on a multi-family property.

Its very easy to get stuck with your ex-spouse's debt or for both parties to get stuck with it, and for the lawyers to advise you to not pay it until the settlement closes because they believe the spouse should pay. 

But in reality most owners with debt dont make much more than 10% on their cash and as a PM in most markets your making 10% with no exposure to loan debt or tenant trash's the unit etc etc.. 

- still no debt or are you considering a cash-out refi now or in the future?

The same if you are open to being an LP on a RE deal, or buying debt, or a laundromat, small business, etc.

Should I aggressively pay down the debt or let the tenants pay the debts off slowly over time?  

The question seems to be, "do I go into debt or downsize"?.

If we pick up this extra $24k, we can pay down variable debt or invest in other properties.However, this obviously is not good for cash flow.

So you end up with a new investment property free and clear of debt or you end up with tax free cash in your pocket and all remaining tax deferred into the new property.  

Leverage is great, and most people my age don't understand the art of good debt, or the fact that there is "good" debt.