If (when) that fund gets depleted I just forgo taking income until it's "full" again.  

Special shout out to @Randall Vanfossen for going over and above with providing me with guidance on how exactly I should go about closing on this deal, and referring me to some local closing attorneys.  

I'm curious as to what others think is the better play: Pay down my loans with the additional cash or forgo that to start saving for a down payment on purchasing a primary residence (which I plan to house hack with either roommates in a SFR or similar with a duplex).

Your infinite% return for the sponsor is the same infinite% return that you get for going to work every day.

Their songs mix with the sound of an old mission style fountain.

Now I have to not only try to get my new purchase fixed while having my family shoved in two rooms with all our stuff in storage for going on MONTHS (we are now around Mar. 2017), but now I have to get my rental ready to show and rented while working a full time job! 

In these cases, I do sometimes counsel landlords that if they are renting to "good" tenants and are not interested in getting educated on the topic or in paying me to assist in the process, perhaps forgo it. 

Keep it simple, forgo unneeded expenses, and get it rented.  

The question you're going to want to ask yourself is "can I really satisfy a 1031 exchange and buy something where the cost of the debt and the reinvestment requirements don't end up costing you more than just paying the tax.Honestly, given the size of the gain vs the size of the property and the subsequent reinvestment target and the amount you'll be able to get out tax free I'd be very tempted to forgo the 1031 and pay the small tax out of the tax free proceeds and move on.

On the one hand, it allowed me to learn the business and set the stage for going out on my own.