The question is, how do I exit? I can't really sell the puts because there isn't
enough open interest in them now that they are so far out of the money.
I have about $150K of funds outside of this position that I could use, but I'm
confused by the rules of exercising a put. Do I have to start shorting the stock?
You certainly don't want to give your broker any instructions to short the stock!
Shorting the stock at this point would actually be increasing your bet that the stock is going to go down more. Worse, a short position in the stock also puts you in a situation of unlimited risk on the stock's upside – a risk you avoided in the first place by using puts. The puts limited your potential loss to only your cost for the options.
There is a scenario where a short position could come into play indirectly, if you aren't careful. If your broker were to permit you to exercise your puts without you having first bought enough underlying shares, then yes, you would end up with a short position in the stock. I say "permit you" because most brokers don't allow clients to take on short positions unless they've applied and been approved for short positions in their account.
In any case, since you are interested in closing out your position and taking your profit, exercising only and thus ending up with a resulting open short position in the underlying is not the right approach. It's not really a correct intermediate step, either.
Rather, you have two typical ways out:
Sell the puts. @quantycuenta has pointed out in his answer that you should be able to sell for no less than the intrinsic value, although you may be leaving a small amount of time value on the table if you aren't careful. My suggestion is to consider using limit orders and test various prices approaching the intrinsic value of the put. Don't use market orders where you'll take any price offered, or you might be sorry.
If you have multiple put contracts, you don't need to sell them all at once. With the kind of profit you're talking about, don't sweat paying a few extra transactions worth of commission.
Exercise the puts. Remember that at the other end of your long put position is one (or more) trader who wrote (created) the put contract in the first place. This trader is obligated to buy your stock from you at the contract price should you choose to exercise your option.
But, in order for you to fulfill your end of the contract when you choose to exercise, you're obligated to deliver the underlying shares in exchange for receiving the option strike price. So, you would first need to buy underlying shares sufficient to exercise at least one of the contracts. Again, you don't need to do this all at once. @PeterGum's answer has described an approach.
(Note that you'll lose any remaining time value in the option if you choose to exercise.)
Finally, I'll suggest that you ought to discuss the timing and apportioning of closing out your position with a qualified tax professional. There are tax implications and, being near the end of the year, there may be an opportunity* to shift some/all of the income into the following tax year to minimize and defer tax due.
* Be careful if your options are near expiry! Options typically expire on the 3rd Friday of the month.
