Illness/Sickness: Sick firebellied newt

Hello. I've got an issue with a newt acting abnormally, and I think he's ill. I keep two male newts in a 10-gallon tank (crappy photo album here: https://imgur.com/a/fbutl) , and I believe they're both over ten years old. I cleaned the tank and changed the water about five days ago. Since then, this little guy has been hiding behind a dry rock in the corner and acting very sluggish. I put food in the tank and it disappears, but I never see him eating. The sick one sits in a dry corner behind a rock for hours at a time and has stopped spending much time in the water. I tried feeding him freeze-dried bloodworms just a bit ago and he snapped at the spoon, however he didn't seem to swallow anything. The temperature in the tank is 66-67 degrees fahrenheit, and they get little to no actual sunlight.

Some quick internet homework suggests he might be impacted, but I don’t know how to be sure of that or how to treat it. Any thoughts?

Hi, welcome to the site!
It could be any number of things, but if you think it is impacted and can't pass the blockage, a good vet is your only option.
I would say it's more likely the conditions in the tank that are upsetting it.
Japanese firebellys like yours will happily live aquatically all year round, so they do best in clean aquariums of at least ten gallons with live plants, a small island or floating cork bark is the only land they need although once settled they'll hardly, if ever use them. Piling up the gravel like that just takes up valuable water volume and traps waste which quickly fouls the water and cause the newts to refuse to enter it
I would do some research and correct their tank and see what happens. As things stand I doubt it will be long before the other one goes off its food as well. Here are some links to get you started.

Caudata Culture Articles - Water Quality

Caudata Culture Species Entry - Cynops pyrrhogaster - Japanese firebelly

Somewhat discouraging, but I'll start there. They lived with a layer of gravel in the tank and less than an inch of water for about ten years with their previous owner before I redid the tank as it is right now. They've had a few gallons of water for about three months without any issues, but I've been paranoid about making too many changes for fear of them not acclamating. I guess I have to learn how to put a tank together.

It' actually very simple to set up a tank for firebellies, just aim for something suitable for cold water fish with live plants and a floating island of cork. I either don't bother with substrate, or use a thin layer of kids play sand. Don't underestimate the importance of live plants, I normally use the cheap bunched stuff they sell in petstores, the newts will spend most of their time there given the choice, and they will help a lot with water quality while the tank is cycling.
If there's anything you're stuck with shoot away and I'll help if I can.