A lower trade in at a dealer could be offset by a sales tax credit (if your state allows) but only applies if you buy another car from the dealer, not lease. You’d have to do the math to see if it makes sense. But if you are dead set on leasing, this is moot, and a private sale would probably help you out with the negative equity. It does take time and effort though.
F150 is in good mechanical condition and low miles however it is a 2006 model.
The $7k cash includes the cash value of the F150 and additional dispensable cash on hand.
We initially planned to wait until January where I could pay the NE and have additional down payment but unfortunately we need something prior to the end of November at the latest.
While yes I could continue driving the F150, it doesn’t solve our issue of needing a vehicle where we can safely travel with baby and dogs.
And yes I understand the negative equity doesn’t go poof and disappear. That’s why we are looking for an SUV that leases well enough that even with the additional NE, we can still be in that $600/month ball park and finally have a vehicle that fits our immediate needs.
I had not understood you planned on selling the F150 to make up the 7k.
I think you’re better off selling the F150 for as close to 7k as you can and then sell the Focus to Carmax, killing the payoff. Start with something fresh, it will give you more flexibility and better options instead of figuring out how to roll over negative equity.
Orthos deal with this all the time. Talk to one and see if he/she has a recommended cushion or pad. Run to any surgical supply shop and try some out.
Keep in mind that youll be paying rent charge on whatever negative equity you roll in, so youll be taking that $7k and making it more.
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Focus ST is somewhat of an enthusiast car. I would think you would be able to do much better selling yourself on FB Marketplace or a Focus forum.
I missed that it’s an ST. 
agreed you’d net more from a private party sale to an enthusiast.
Just to be clear, this means you are looking for a mid/large SUV for under $300/mo as your NE itself is going to be at least $300/mo assuming you don’t have a high money factor.
There are exceedingly few vehicles that meet that criteria and can absorb $11k of NE.
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Like others have said, you could definitely do better than 10k selling the ST privately. It would make a great first-ish car for a budding car enthusiast if it’s still in good shape - FBMP and Craigslist are great places to sell.