Esophagus closing too quickly? Has the infection reached my brain?

No, it's that it's officially hit your nervous system - which is bad enough. When Lyme attacks the nervous system, it damages the myelin sheath - the protective covering over the nerve itself. So messages from the brain to whatever part of the body is affected gets mixed up signals. You are trying to swallow, and your throat is trying to breath kind of thing.

While many of us have these issues with Neuro Lyme, not everyone has it in their throat and this could potentially be dangerous, as you could choke. So I encourage you to continue to look for answers to help as you go through treatments - as that's the only permanent fix for this.

The myelin sheath will regrow once the infections are taken care of. I've had a lot of improvements with my nervous system once I healed my infections.

You can try cutting your food into very small pieces, keeping water near you as you eat so you can take a small sip of water as needed, sticking to food that has a lot of moisture in it (not dry like a roasted meat, but like a stew or casserole), being sure to chew your food very, very thoroughly and other things like that.

I also had the narcolepsy, but it also resolved with treatments.

I hope that you can find some good "work arounds" for you as you go through treatments.

Swallowing is a very complicated coordinated movement that involves the cranial nerves particularly the vagus and the glossopharyngeal. The glossopharyngeal is given most of the credit but how far down you are having issues sounds more like the vagus nerve. They both originate off of your brain stem and many people have issues with swallowing after having a stroke. Not saying that this is what happened to you but if your brain and brain stem are being damaged/inflamed by lyme then this is a possible symptom, maybe get an MRI if you haven't. Does your uvula deviate, if so there is a strong probability of being neurological.
I would second the idea of a sleep study, you have a lot of other issues that need to be ruled out like possible causes of the narcolepsy, hallucinations and sleep paralysis. My sleep paralysis completely went away with CPAP, it didn't do much for the drowsy feeling however.

I had an MRI done and everything was normal. No signs of stroke or ms or tumor. Which is what they were initially looking for. Everything came back normal though thank god. Although sometimes I wish they'd find something just so I could treat it. Starting the Cowden protocol as soon as I can find a doctor to sign off on it. I can't afford it otherwise. Good suggestion on the nervous system. Guess I'll pencil that appointment in before the end of the year too. Thankfully we've hit our deductible so I'm getting all these specialists in. My sleep study was positive for narc I hope to god it goes away too.

I am much like Traveler said, I cannot take a bite of roasted meat. It gets stuck and I have to coach it down with water. It is scary. Now I lube my throat up with water before I take a TEENY bite of any meat. And I am cautious with all other foods.

Lyme messes with most everything.